I am have recently bought a dual core processor,
I don't even know what it is able to do,

Games like planetside 2 are easily played without lag,
I had just one lag of 1 and half second.

without telling of RAM memory 4gb and windows 7 64 bits.

fast and no crashes and anything like that.

I kinda don't understand.

It does things like a normal processor would, but on 2 cores.

well, I was just asking for orientation of what could use a good part of dual core,
It is a little reason for multiple cores,

I know, it is faster and with two cores, but I dont know really why this is implemented.

Imagine you wanted to eat a lot in a little amount of time. It would be easier with 2 mouths wouldn't it?

I think I know what you are actually asking, and well... there are a lot of softwares that use two cores(or even more), your operating system also uses more than 1 core.

Playing crysis used all 4 of mine. Most programs automatically utilize all the cores.

The OS will automatically assign threads and processes to cores as necessary, you don't need to do anything.

Anything extremely CPU intense should max out your cores, for example I max it out using Evolution Chamber which is a build order optimizer for SC2.