user8646: I have got this library+code example from a friend, to read RFID numbers using the RFID-RC522 chip:
RFID.ino
#include <RFID.h>
#include <SPI.h>

/**
* RFID pins
* MOSI: Pin 11 / ICSP-4
* MISO: Pin 12 / ICSP-1
* SCK: Pin 13 / ISCP-3
* SS: Pin 10
* RST: Pin 9
*/

RFID rfid(10, 9);
unsigned char prevNum[] = {0,0,0,0,0};

void setup()
{
  Serial.begin(9600);
  SPI.begin();
  rfid.init();
}

void loop() {
    if (rfid.isCard()){
        if (rfid.readCardSerial() && (
            rfid.serNum[0] != prevNum[0]
         or rfid.serNum[1] != prevNum[1]
         or rfid.serNum[2] != prevNum[2]
         or rfid.serNum[3] != prevNum[3]
         or rfid.serNum[4] != prevNum[4]
        )){
          Serial.println(rfid.serNum[0]);
          Serial.println(rfid.serNum[1]);
          Serial.println(rfid.serNum[2]);
          Serial.println(rfid.serNum[3]);
          Serial.println(rfid.serNum[4]);

          prevNum[0] = rfid.serNum[0];
          prevNum[1] = rfid.serNum[1];
          prevNum[2] = rfid.serNum[2];
          prevNum[3] = rfid.serNum[3];
          prevNum[4] = rfid.serNum[4];
        }
    }
}

RFID.h
/* RFID.h - Library to use ARDUINO RFID MODULE KIT 13.56 MHZ WITH TAGS SPI W AND R BY COOQROBOT.
 * Based on code Dr.Leong   ( WWW.B2CQSHOP.COM )
 * Created by Miguel Balboa (circuitito.com), Jan, 2012.
 */
#ifndef RFID_h
#define RFID_h

#include <Arduino.h>
#include <SPI.h>



/******************************************************************************
 * Definitions
 ******************************************************************************/
#define MAX_LEN 16   // Largo m�ximo de la matriz

//MF522 comando palabra
#define PCD_IDLE              0x00               // NO action; Y cancelar el comando
#define PCD_AUTHENT           0x0E               // autenticaci�n de clave
#define PCD_RECEIVE           0x08               // recepci�n de datos
#define PCD_TRANSMIT          0x04               // Enviar datos
#define PCD_TRANSCEIVE        0x0C               // Enviar y recibir datos
#define PCD_RESETPHASE        0x0F               // reajustar
#define PCD_CALCCRC           0x03               // CRC calcular

//Mifare_One  Tarjeta Mifare_One comando palabra
#define PICC_REQIDL           0x26               // �rea de la antena no est� tratando de entrar en el estado de reposo
#define PICC_REQALL           0x52               // Todas las cartas para encontrar el �rea de la antena
#define PICC_ANTICOLL         0x93               // anti-colisi�n
#define PICC_SElECTTAG        0x93               // elecci�n de tarjeta
#define PICC_AUTHENT1A        0x60               // verificaci�n key A
#define PICC_AUTHENT1B        0x61               // verificaci�n Key B
#define PICC_READ             0x30               // leer bloque
#define PICC_WRITE            0xA0               // Escribir en el bloque
#define PICC_DECREMENT        0xC0               // cargo
#define PICC_INCREMENT        0xC1               // recargar
#define PICC_RESTORE          0xC2               // Transferencia de datos de bloque de buffer
#define PICC_TRANSFER         0xB0               // Guardar los datos en el b�fer
#define PICC_HALT             0x50               // inactividad

//MF522 C�digo de error de comunicaci�n cuando regres�
#define MI_OK                 0
#define MI_NOTAGERR           1
#define MI_ERR                2

//------------------ MFRC522 registro---------------
//Page 0:Command and Status
#define     Reserved00            0x00
#define     CommandReg            0x01
#define     CommIEnReg            0x02
#define     DivlEnReg             0x03
#define     CommIrqReg            0x04
#define     DivIrqReg             0x05
#define     ErrorReg              0x06
#define     Status1Reg            0x07
#define     Status2Reg            0x08
#define     FIFODataReg           0x09
#define     FIFOLevelReg          0x0A
#define     WaterLevelReg         0x0B
#define     ControlReg            0x0C
#define     BitFramingReg         0x0D
#define     CollReg               0x0E
#define     Reserved01            0x0F
//Page 1:Command
#define     Reserved10            0x10
#define     ModeReg               0x11
#define     TxModeReg             0x12
#define     RxModeReg             0x13
#define     TxControlReg          0x14
#define     TxAutoReg             0x15
#define     TxSelReg              0x16
#define     RxSelReg              0x17
#define     RxThresholdReg        0x18
#define     DemodReg              0x19
#define     Reserved11            0x1A
#define     Reserved12            0x1B
#define     MifareReg             0x1C
#define     Reserved13            0x1D
#define     Reserved14            0x1E
#define     SerialSpeedReg        0x1F
//Page 2:CFG
#define     Reserved20            0x20
#define     CRCResultRegM         0x21
#define     CRCResultRegL         0x22
#define     Reserved21            0x23
#define     ModWidthReg           0x24
#define     Reserved22            0x25
#define     RFCfgReg              0x26
#define     GsNReg                0x27
#define     CWGsPReg              0x28
#define     ModGsPReg             0x29
#define     TModeReg              0x2A
#define     TPrescalerReg         0x2B
#define     TReloadRegH           0x2C
#define     TReloadRegL           0x2D
#define     TCounterValueRegH     0x2E
#define     TCounterValueRegL     0x2F
//Page 3:TestRegister
#define     Reserved30            0x30
#define     TestSel1Reg           0x31
#define     TestSel2Reg           0x32
#define     TestPinEnReg          0x33
#define     TestPinValueReg       0x34
#define     TestBusReg            0x35
#define     AutoTestReg           0x36
#define     VersionReg            0x37
#define     AnalogTestReg         0x38
#define     TestDAC1Reg           0x39
#define     TestDAC2Reg           0x3A
#define     TestADCReg            0x3B
#define     Reserved31            0x3C
#define     Reserved32            0x3D
#define     Reserved33            0x3E
#define     Reserved34                0x3F
//-----------------------------------------------

class RFID
{
  public:
    RFID(int chipSelectPin, int NRSTPD);

    bool isCard();
    bool readCardSerial();

    void init();
    void reset();
    void writeMFRC522(unsigned char addr, unsigned char val);
    void antennaOn(void);
    unsigned char readMFRC522(unsigned char addr);
    void setBitMask(unsigned char reg, unsigned char mask);
    void clearBitMask(unsigned char reg, unsigned char mask);
    void calculateCRC(unsigned char *pIndata, unsigned char len, unsigned char *pOutData);
    unsigned char MFRC522Request(unsigned char reqMode, unsigned char *TagType);
    unsigned char MFRC522ToCard(unsigned char command, unsigned char *sendData, unsigned char sendLen, unsigned char *backData, unsigned int *backLen);
    unsigned char anticoll(unsigned char *serNum);
    unsigned char auth(unsigned char authMode, unsigned char BlockAddr, unsigned char *Sectorkey, unsigned char *serNum);
    unsigned char read(unsigned char blockAddr, unsigned char *recvData);
    unsigned char write(unsigned char blockAddr, unsigned char *writeData);
    void halt();

    unsigned char serNum[5];       // Constante para guardar el numero de serie leido.
    unsigned char AserNum[5];      // Constante para guardar el numero d serie de la secion actual.

  private:
    int _chipSelectPin;
    int _NRSTPD;

};

#endif

Using the code I have now I can read RFID data, and read them via the Serial monitor.
Now, if I want to write RFID numbers to a card, can I do this with this library, or what other library should I use that is capable of writing?

user32250: The library you refer to is called RFID by Miguel Balboa. 
It is equipped with several examples showing you how to read and write data. Check them out.
If you are looking for some more background information on what happens when you write data on a tag, check out the first video of of the MakeCourse of the University of South Florida. 
At 21:30 it explains how to write data.

8646: <p>I have got this library+code example from a friend, to read RFID numbers using the RFID-RC522 chip:</p>

<p>RFID.ino</p>

<pre><code>#include &lt;RFID.h&gt;
#include &lt;SPI.h&gt;

/**
* RFID pins
* MOSI: Pin 11 / ICSP-4
* MISO: Pin 12 / ICSP-1
* SCK: Pin 13 / ISCP-3
* SS: Pin 10
* RST: Pin 9
*/

RFID rfid(10, 9);
unsigned char prevNum[] = {0,0,0,0,0};

void setup()
{
  Serial.begin(9600);
  SPI.begin();
  rfid.init();
}

void loop() {
    if (rfid.isCard()){
        if (rfid.readCardSerial() &amp;&amp; (
            rfid.serNum[0] != prevNum[0]
         or rfid.serNum[1] != prevNum[1]
         or rfid.serNum[2] != prevNum[2]
         or rfid.serNum[3] != prevNum[3]
         or rfid.serNum[4] != prevNum[4]
        )){
          Serial.println(rfid.serNum[0]);
          Serial.println(rfid.serNum[1]);
          Serial.println(rfid.serNum[2]);
          Serial.println(rfid.serNum[3]);
          Serial.println(rfid.serNum[4]);

          prevNum[0] = rfid.serNum[0];
          prevNum[1] = rfid.serNum[1];
          prevNum[2] = rfid.serNum[2];
          prevNum[3] = rfid.serNum[3];
          prevNum[4] = rfid.serNum[4];
        }
    }
}
</code></pre>

<p>RFID.h</p>

<pre><code>/* RFID.h - Library to use ARDUINO RFID MODULE KIT 13.56 MHZ WITH TAGS SPI W AND R BY COOQROBOT.
 * Based on code Dr.Leong   ( WWW.B2CQSHOP.COM )
 * Created by Miguel Balboa (circuitito.com), Jan, 2012.
 */
#ifndef RFID_h
#define RFID_h

#include &lt;Arduino.h&gt;
#include &lt;SPI.h&gt;



/******************************************************************************
 * Definitions
 ******************************************************************************/
#define MAX_LEN 16   // Largo m�ximo de la matriz

//MF522 comando palabra
#define PCD_IDLE              0x00               // NO action; Y cancelar el comando
#define PCD_AUTHENT           0x0E               // autenticaci�n de clave
#define PCD_RECEIVE           0x08               // recepci�n de datos
#define PCD_TRANSMIT          0x04               // Enviar datos
#define PCD_TRANSCEIVE        0x0C               // Enviar y recibir datos
#define PCD_RESETPHASE        0x0F               // reajustar
#define PCD_CALCCRC           0x03               // CRC calcular

//Mifare_One  Tarjeta Mifare_One comando palabra
#define PICC_REQIDL           0x26               // �rea de la antena no est� tratando de entrar en el estado de reposo
#define PICC_REQALL           0x52               // Todas las cartas para encontrar el �rea de la antena
#define PICC_ANTICOLL         0x93               // anti-colisi�n
#define PICC_SElECTTAG        0x93               // elecci�n de tarjeta
#define PICC_AUTHENT1A        0x60               // verificaci�n key A
#define PICC_AUTHENT1B        0x61               // verificaci�n Key B
#define PICC_READ             0x30               // leer bloque
#define PICC_WRITE            0xA0               // Escribir en el bloque
#define PICC_DECREMENT        0xC0               // cargo
#define PICC_INCREMENT        0xC1               // recargar
#define PICC_RESTORE          0xC2               // Transferencia de datos de bloque de buffer
#define PICC_TRANSFER         0xB0               // Guardar los datos en el b�fer
#define PICC_HALT             0x50               // inactividad

//MF522 C�digo de error de comunicaci�n cuando regres�
#define MI_OK                 0
#define MI_NOTAGERR           1
#define MI_ERR                2

//------------------ MFRC522 registro---------------
//Page 0:Command and Status
#define     Reserved00            0x00
#define     CommandReg            0x01
#define     CommIEnReg            0x02
#define     DivlEnReg             0x03
#define     CommIrqReg            0x04
#define     DivIrqReg             0x05
#define     ErrorReg              0x06
#define     Status1Reg            0x07
#define     Status2Reg            0x08
#define     FIFODataReg           0x09
#define     FIFOLevelReg          0x0A
#define     WaterLevelReg         0x0B
#define     ControlReg            0x0C
#define     BitFramingReg         0x0D
#define     CollReg               0x0E
#define     Reserved01            0x0F
//Page 1:Command
#define     Reserved10            0x10
#define     ModeReg               0x11
#define     TxModeReg             0x12
#define     RxModeReg             0x13
#define     TxControlReg          0x14
#define     TxAutoReg             0x15
#define     TxSelReg              0x16
#define     RxSelReg              0x17
#define     RxThresholdReg        0x18
#define     DemodReg              0x19
#define     Reserved11            0x1A
#define     Reserved12            0x1B
#define     MifareReg             0x1C
#define     Reserved13            0x1D
#define     Reserved14            0x1E
#define     SerialSpeedReg        0x1F
//Page 2:CFG
#define     Reserved20            0x20
#define     CRCResultRegM         0x21
#define     CRCResultRegL         0x22
#define     Reserved21            0x23
#define     ModWidthReg           0x24
#define     Reserved22            0x25
#define     RFCfgReg              0x26
#define     GsNReg                0x27
#define     CWGsPReg              0x28
#define     ModGsPReg             0x29
#define     TModeReg              0x2A
#define     TPrescalerReg         0x2B
#define     TReloadRegH           0x2C
#define     TReloadRegL           0x2D
#define     TCounterValueRegH     0x2E
#define     TCounterValueRegL     0x2F
//Page 3:TestRegister
#define     Reserved30            0x30
#define     TestSel1Reg           0x31
#define     TestSel2Reg           0x32
#define     TestPinEnReg          0x33
#define     TestPinValueReg       0x34
#define     TestBusReg            0x35
#define     AutoTestReg           0x36
#define     VersionReg            0x37
#define     AnalogTestReg         0x38
#define     TestDAC1Reg           0x39
#define     TestDAC2Reg           0x3A
#define     TestADCReg            0x3B
#define     Reserved31            0x3C
#define     Reserved32            0x3D
#define     Reserved33            0x3E
#define     Reserved34                0x3F
//-----------------------------------------------

class RFID
{
  public:
    RFID(int chipSelectPin, int NRSTPD);

    bool isCard();
    bool readCardSerial();

    void init();
    void reset();
    void writeMFRC522(unsigned char addr, unsigned char val);
    void antennaOn(void);
    unsigned char readMFRC522(unsigned char addr);
    void setBitMask(unsigned char reg, unsigned char mask);
    void clearBitMask(unsigned char reg, unsigned char mask);
    void calculateCRC(unsigned char *pIndata, unsigned char len, unsigned char *pOutData);
    unsigned char MFRC522Request(unsigned char reqMode, unsigned char *TagType);
    unsigned char MFRC522ToCard(unsigned char command, unsigned char *sendData, unsigned char sendLen, unsigned char *backData, unsigned int *backLen);
    unsigned char anticoll(unsigned char *serNum);
    unsigned char auth(unsigned char authMode, unsigned char BlockAddr, unsigned char *Sectorkey, unsigned char *serNum);
    unsigned char read(unsigned char blockAddr, unsigned char *recvData);
    unsigned char write(unsigned char blockAddr, unsigned char *writeData);
    void halt();

    unsigned char serNum[5];       // Constante para guardar el numero de serie leido.
    unsigned char AserNum[5];      // Constante para guardar el numero d serie de la secion actual.

  private:
    int _chipSelectPin;
    int _NRSTPD;

};

#endif
</code></pre>

<p>Using the code I have now I can read RFID data, and read them via the Serial monitor.
Now, if I want to write RFID numbers to a card, can I do this with this library, or what other library should I use that is capable of writing?</p>
 32250: <p>The library you refer to is called <a href="https://github.com/miguelbalboa/rfid" rel="nofollow noreferrer">RFID</a> by Miguel Balboa. 
It is equipped with several examples showing you how to read and write data. Check them out.</p>

<p>If you are looking for some more background information on what happens when you write data on a tag, check out the <a href="http://makecourse.weebly.com/week10segment1.html" rel="nofollow noreferrer">first video of of the MakeCourse</a> of the University of South Florida. </p>

<p>At 21:30 it explains how to write data.</p>
