How Vectors work and how I should use them?

I would really appreciate if someone could explain to me how vector 2 and vector 3 works, especially the vector 2 as i’m doing a 2d Platform Game.
I’ve already seen the documentation and have some doubts.

How I should use Vectors for movement for example?
How it works?

They’re basically the same, for two dimensions and three dimensions respectively. This article covers the basics:

http://www.gamedev.net/page/resources/_/technical/math-and-physics/practical-use-of-vector-math-in-games-r2968

It’s well worth getting your head round it properly, as this is fundamental to all spatial modelling.