A subset is just containment, it is just a collection of points inside the vector space. A subspace is both a subset and a vector space in itself, that is it is closed under addition and multiplication by the field
Let S be a subset of a vector space V.
The span S is a subspace of V which contains S...
think of the defenition of subset, i.e in a language that you an i can understand, a subset are a set of elements (in its on right) that belong to the vector space.
A vector space you should read your book to understand the properties, im assuming that is for a linear class, dont worry you will most like just need to understand what is a subspace...
a subspace has to be a subset in the vector space, that is has to be elements in the vector space, this subset s=list of elements (v1,...vn), the span(s)={a1v1+...+anvn:a1,..an are elements in a the field}, apply three properties of the vector space,the additive id, closed under addition and close under scalar multiplication.. if this three hold then span is a subspace of the vector space, ( is true you must try)