without getting too technical here are a few thoughts which may help. if not, my apologies!
first be aware of two of the persistent problems with math terminology, which can hinder progress in the early stages of study.
- ambiguity - same word used in different senses. one recurrent offender is normal
- words used which are common in everyday language, but which have a much more specific meaning that may be quite different e.g. group, field, ring, root, space.
these everyday meanings may in some cases be helpful, but can also lead to confusion
a space is a set with additional structure. if a topology is specified we have a topological space. if, on the other hand, the set has a linear structure (which is algebraic in nature), it is called a linear space or in some cases a vector space. if we have both structures then the object is a topological vector space. a topological group is a set which has both a group structure and a linear structure which are (in a specific way) compatible. a vector space is an abelian group, but with further structure relating to multiplication of vector by a scalar. the scalars belong to a field associated with a vector space. if the vector space is $V$ and the field is $F$ then we say $V$ is a vector space over $F$.
science students are programmed early to think of a vector as an almost physical thing. arrows with three co-ordinates and three different kinds of 'multiplication' - two of which are rather unfamiliar and one of which (the so-called cross product) seems initially to be deliberately obscure. whilst 3-D vectors over the real numbers (the field $\mathbb{R})$ form a perfectly good vector space, one should try to not to let this specific example monopolize intuition. for example every field is a vector space over itself.
let $F$ be a field and let $X$ be any non-empty set. there is a well-defined set $F^X$ whose elements are all the functions from $X$ to $F$. even though $X$ may have no algebraic structure at all it is clear that if $f,g \in F^X$ we can add these functions together by adding their images in $F$ and the result will be another function from $X$ to $F$. likewise a function can be multiplied by a scalar in $F$ to give another function. here the functions are the vectors of a vector space whose associated field is $F$. actually this construction is always there, perhaps in the backkground, obscured by some more concrete intuition..
so, for example, if $X$ is the set $\{1,2,\dots,n\}$ and $X$ is the field of real numbers $\mathbb{R}$ then we obtain the familiar vector space $\mathbb{R}^n$
in simplistic terms you may think of a subspace as a subset of the underlying set of the space, which satisfies the same defining axioms as the space itself. for a vector space this means that (i) for any vector in the subset, all its scalar multiples are also in the subset (ii) for any two vectors in the subset, their sum is also in the subset.
zero must be in the subspace because $-1$ is in the field, and thus if $v$ is in the subspace so is $-v$ (abbreviation for $(-1)v$). since the sum of any two subspace vectors must be in the subspace so must $v-v=0$ (remember a vector space is an abelian group, and this is therefore also required of a subspace)