Q:

how to unerstand finite dimensional vector spaces

A vector space $V$ is called finite-dimensional if there is a finite subset of $V$ that is a basis for $V$. If there is no such finite subset of $V$, then $V$ is called infinite-dimensional. 

We now establish some results about finite-dimensional vector spaces that will tell about the number of vectors in a basis, compare two different bases, and give properties of bases.  First, we observe that if $\{\mathbf{v}_1, \mathbf v_2,\dotsc, \mathbf v_k\}$ is a basis for a vector space $V$, then $\{c\mathbf v_1, \mathbf v_2, \dotsc, \mathbf v_k\}$ is also a basis when $c\neq 0$ (Exercise 35).  Thus a basis for a nonzero vector space is never unique. 
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I am confused if $\mathbb R^2$ is a finite dimensional vector space.  $[1 \ 0],[0 \  1]$ will be the standard basis of $\mathbb R^2$. However, there are also $[2 \ 0],[0 \ 1]$ and I can find infinite many to be the basis of $\mathbb R^2$. So, $\mathbb R^2$ is an infinite dimensional vector space?   

A:

I suppose you are mixing the terms "finite subset" and "finite number of subset". 
The first one corresponds to the fact that in the subset there are only finitely many ${\it elements}$; the second one corresponds to the fact that there are finitely many ${\it subsets}$.
Hope I helped.  I also mixed these two concepts at the start of learning linear algebra, so I think you met the same problem.