user2966: I have an Arduino Ethernet shield. 
May I connect it to a machine that´s connected wirelessly to the internet in order to get the arduino into the internet? 
Or do I need to connect it only to a router?

user70: Assuming an Ethernet port on the PC, the Ethernet shield can be connected to it using an Ethernet crossover cable:

(source: Wikipedia)
Many recent PC network interfaces have automatic cross-over (Auto MDI-X), with which even a regular (straight) Ethernet patch cable will work. 
This merely addresses the physical Ethernet connectivity. 
For actually routing IP packets between the Arduino and the rest of the Internet, some software router or software bridge is required to run on the PC. Several open source routing products exist, especially for Linux and FreeBSD, but that is outside the scope of this StackExchange.
Finally, the Ethernet shield does not need a router, it an as well be hooked up to an Ethernet switch or hub. 

2966: <p>I have an Arduino Ethernet shield. </p>

<p>May I <strong>connect it to a machine that´s connected wirelessly to the internet</strong> in order to get the arduino into the internet? </p>

<p>Or do I need to connect it <strong>only</strong> to a router?</p>
 70: <p>Assuming an Ethernet port on the PC, the Ethernet shield can be connected to it using an <a href="http://en.wikipedia.org/wiki/Ethernet_crossover_cable" rel="nofollow noreferrer">Ethernet <strong>crossover</strong> cable</a>:</p>

<p><img src="https://i.stack.imgur.com/xbJ3g.png" alt="enter image description here">
(<em>source: <a href="http://en.wikipedia.org/wiki/File:Ethernet_MDI_crossover.svg" rel="nofollow noreferrer">Wikipedia</a></em>)</p>

<p>Many recent PC network interfaces have automatic cross-over (<a href="http://en.wikipedia.org/wiki/Medium_Dependent_Interface#Auto_MDI-X" rel="nofollow noreferrer">Auto MDI-X</a>), with which even a regular (straight) Ethernet patch cable will work. </p>

<p>This merely addresses the <strong>physical Ethernet connectivity</strong>. </p>

<p>For <strong>actually routing IP packets</strong> between the Arduino and the rest of the Internet, some software router or software bridge is required to run on the PC. Several open source routing products exist, especially for Linux and FreeBSD, but that is outside the scope of this StackExchange.</p>

<p>Finally, the Ethernet shield does not need a router, it an as well be hooked up to an Ethernet <strong>switch</strong> or <strong>hub</strong>. </p>
