Vector spaces, Spans and linear independence

Vector spaces, Spans and linear independence...?

I'm just having a little bit of trouble in my Linear Algebra class and am wondering if someone can explain some of the above listed things so that a layman can understand them...

Just to give you an idea of what I'm confused about:
-I understand that a vector space is some mathematical construct defined by what closed operations can be done on it...but is there some physical analogy or something as to what exactly a vector space is or why they're useful?

- I don't get how to find a Span or "spanning set" or what this "Span" actually represents - any information would be helpful :)

- I understand what linear independence is and how to find it given a set of vectors, but I was given this problem and have no idea where to start:
*Let V be a vector space, W be a subspace of V, S a linearly independent subset of W, and v ? V - W. Prove that S U {v} is linearly independent.

Any help understanding this will be much appreciated :)