Intuition/motivation/why should I care about (definition of) support 
Let $X$ be a metric space. If $f: X \to \mathbb{R}$, the support of $f$ is the closure of the set $\{x: f(x) \neq 0\}$.

What is the intuition/motivation/why should I care about the (definition of) support?
 A: The support of a function can be of significance in the study of differentiable manifolds. For example, Stokes' theorem (a generalization of theorems from multivariable calculus such as Green's theorem) requires the differential form to have compact support. 
Another example of support is in the definition of a bump function, an infinitely differentiable function with compact support. These allow us to make partitions of unity, which allow us to define the value of an integral on a surface even though the charts (the maps from the surface to the Euclidean space, which define the surface) are only locally defined. 
A: The answer by Florence is sufficient, but to provide another reason, take for instance $C(\mathbb{R})$, so the ring of continuous functions defined on $\mathbb{R}$. The functions of compact support form an ideal in this space. This is interesting because it provides a counterexample to the claim:
"Given a maximal ideal $I$ in $C(\mathbb{R})$, $I= \mathfrak{m}_p$ for some $p \in \mathbb{R}$, where $\mathfrak{m}_p= \left\{ f \in C(\mathbb{R}): f(p)=0 \right\}$."
So this claim is not true. However, if we add the condition that the residue field of $I$ is $\mathbb{R}$, the claim becomes true. So then this begs the question: what is the residue field of $I$ if $I$ is the maximal ideal containing all functions of compact support? I asked this question a while ago and essentially got the answer: "you probably won't be able to describe such a thing." I think this is very interesting in the sense that we can use ideals of functions of compact support to assert the existence of fields we probably can't even describe!
