2D and 3D

2D means 2-dimensional meaning there is only a height and a width.
2D is typically used in old arcade games where the graphics weren't very good because the hardware wasn't as advanced as today. Most older games where made in 2D because it was easier, faster and cheaper.
The platforms that used 2D where some PS1 games and handheld games consoles such as the Gameboy Advance SP. Some famous 2D games are Pong and the early Mario and Sonic games.

3D is 3-dimensional, it has height and width but also depth and this enables players to move into the screen and look around objects.
3D is used in more modern games this means better quality games. However making 3D games is slower and more complex, meaning that the games usually cost more money to make and so more money to purchase.
3D games are made for the modern games consoles such as Xbox 360's, PS3's and  PC's.
Some famous 3D games are Call of Duty 4 and Halo 3.