I was just wondering if I were to buy an Arduino Mega and an Ethernet shield, and i were to connect the Ethernet shield to a router, if i'd be able to send commands to the arduino from a computer on the same router, almost like one does with serial commands.
Don't be put off by the "extra" stuff there about "talking to" an Arduino across the internet. What you want to do... talk to across your LAN... is the foundation of the "across the internet" stuff.
You would set your Arduino up at a fixed IP address on the LAN, say 192.168.0.250
You need to tell your router NOT to issue that address to anyone else. (You do this by tweaking its DCHP services, setting aside at least ... 250 as one it doesn't use.)
You have to be sure nothing else on the LAN has that IP address.
Then you set the Arduino up as a server.
You then access it across the LAN just by putting the IP address into a browser (e.g. Firefox) in the address bar... the place you usually put a "www- thingie".
I haven't forgotten the "commands" part of your question. Once you have the Arduino serving up a little "Hello World" page, you'll be 3/4s of the way there. Details of command processing at Arduserve.com... but "walk" first. All of the "simple" stuff above has to be got working before you can do the next stuff.
a long time back I did some tinkering (below) where servo commands were sent from a web page to an arduino with ethernet shield, and the arduino sent the commands out the serial pins to an ssc-32 servo controller.