The 2 by 2 by 2 cube is called a "pocket cube" or a "2 by 2" cube.
A lot of people see and say "wow, that looks really easy!" It isn't. It's the same difficulty as solving the corners of a 3 by 3 cube. Actually, it's a touch harder than that because on a 3 by 3 you can use the centres for colour reference.
The model I have is an Eastsheen mechanism and I would definitely recommend it, it's very sturdy.