Why we care about these two Banach spaces $$C_0(\Omega):=\{ f \in C(\Omega) \colon \forall \epsilon>0 \text{ there exists a compact } K_\epsilon \subset \Omega \text{ such that }\\|f(s)|<\epsilon, \forall s \in \Omega \setminus K_\epsilon\}$$ and $$C_c(\Omega):= \{f \in C(\Omega) \colon\text{ the support of $f$ is compact}\}$$ with the sup-norm. Where $\Omega$ is a locally compact space, and $C(\Omega)$ is the set of all continuous functions in $\Omega$.
What is the intuition behind these definitions?
$C_0(X)$ is the most canonical Banach algebra (actually a $ C^*$-algebra) you may associate to the locally compact space $X$, so much so that the correspondence $$X\to C_0(X)$$ is an equivalence of categories from locally compact spaces to commutative C*-algebras.
Regarding $C_c(X)$, it is among the smallest dense subalgebras of $C_0(C)$, so it is often used as a more manageable gadget in concrete computations.