Compare i3 and core 2 duo

Depends--what do you want to use it for? And what core 2 duo specifically?
The i3 is a better processor for applications such as photoshop. However, in game performance it can lose out to a more powerful core 2 duo. Gaming systems tend to use either high-powered core 2 duos or core i5s, so a system with an i3 will likely not have as good a graphics card.
That should give you a general idea, but to answer completely we need to know what you're looking at and what you want to use it for.

what is better for gaming? is it core 2 duo? which 1 has good graphics and speed?

As I said, it depends. If it's for gaming, I'd look for a Core 2 Duo with a high clock speed, that's probably better. However, a high clocked i3 could work too. Speed will all depend on the clock speed. I can't really say more than that.