user26038: 
Is there any way I can program an Arduino using another Arduino,without the need of any computer. I want to store the programming code in the EEPROM of the "Programmer Arduino" and using any interface like SPI, I2C or Serial, burn this code to the Second arduino. Is it possible? How can it be implemented ?

user29158: Yes. Connect 2 arduinos via serial connection (https://www.arduino.cc/en/Reference/Serial), restart one of them via RESET channel and upload sketch(https://www.arduino.cc/en/Guide/ArduinoProMini#toc1). You will need binary files - google 'arduino get sketch binary'(cannot add any more links).

user10794: It is certainly possible to do this.

I want to store the programming code in the EEPROM of the "Programmer Arduino" and using any interface like SPI, I2C or Serial, burn this code to the Second arduino. 

I should point out that the EEPROM may not be large enough for this. For example, the Uno has only 1KB of EEPROM.
A more workable solution would be to store a reasonable size program in PROGMEM, or a larger one in external memory, like an SD card.
I have a page about a stand-alone programmer - the first idea presented just uses the Serial monitor to command the programmer behaviour:

Using the Serial Monitor you select the file you want programmed. Another approach, commissioned by someone who wanted to eliminate the external computer altogether looks like this:

In this case you put one or more hex files onto the SD card, choose one using the "file select" knob, and then press the "program" button to program it. This only takes a couple of seconds.
There are also commercial products that do a similar thing.

26038: <p><a href="https://i.stack.imgur.com/0az7i.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/0az7i.png" alt="enter image description here" /></a></p>
<p>Is there any way I can program an Arduino using another Arduino,without the need of any computer. I want to store the programming code in the EEPROM of the &quot;Programmer Arduino&quot; and using any interface like SPI, I2C or Serial, <strong>burn</strong> this code to the Second arduino. Is it possible? How can it be implemented ?</p>
 29158: <p>Yes. Connect 2 arduinos via serial connection (<a href="https://www.arduino.cc/en/Reference/Serial" rel="nofollow noreferrer">https://www.arduino.cc/en/Reference/Serial</a>), restart one of them via RESET channel and upload sketch(<a href="https://www.arduino.cc/en/Guide/ArduinoProMini#toc1" rel="nofollow noreferrer">https://www.arduino.cc/en/Guide/ArduinoProMini#toc1</a>). You will need binary files - google 'arduino get sketch binary'(cannot add any more links).</p>
 10794: <p>It is certainly possible to do this.</p>

<blockquote>
  <p>I want to store the programming code in the EEPROM of the "Programmer Arduino" and using any interface like SPI, I2C or Serial, burn this code to the Second arduino. </p>
</blockquote>

<p>I should point out that the EEPROM may not be large enough for this. For example, the Uno has only 1KB of EEPROM.</p>

<p>A more workable solution would be to store a reasonable size program in PROGMEM, or a larger one in external memory, like an SD card.</p>

<p>I have a <a href="http://www.gammon.com.au/forum/?id=11638" rel="nofollow noreferrer">page about a stand-alone programmer</a> - the first idea presented just uses the Serial monitor to command the programmer behaviour:</p>

<p><a href="https://i.stack.imgur.com/DsZ4I.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DsZ4I.jpg" alt="Board programmer"></a></p>

<p>Using the Serial Monitor you select the file you want programmed. Another approach, commissioned by someone who wanted to eliminate the external computer altogether looks like this:</p>

<p><a href="https://i.stack.imgur.com/w3j9f.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/w3j9f.jpg" alt="Stand-alone programmer"></a></p>

<p>In this case you put one or more hex files onto the SD card, choose one using the "file select" knob, and then press the "program" button to program it. This only takes a couple of seconds.</p>

<p>There are also commercial products that do a similar thing.</p>
