What are the differences among an affine variety, a vector space, and a projective variety?

Are there some nice examples to explain this? 

Edit: For example, what is the difference between the following: $k^n$ ($k$ is the ground field) as a vector space, and $k^n$ as an affine variety (what are the equations satisfied by $k^n$)? Why is $k^n$ *not* a projective variety? 

Why do we need to introduce projective varieties (do they have some advantages over affine varieties)? 

Why are $GL\_n$ and $SL\_n$ algebraic varieties? (What equations are satisfied by $GL\_n$ and $SL\_n$)?  
Are $GL\_n$ and $SL\_n$ projective varieties? 

Can some variety be both affine and projective? 

Thank you.