Let $M^n$ be a smooth Manifold, of dimension $n$.
From [Kriegl, Andreas; Michor, Peter W., The convenient setting of global analysis, Mathematical Surveys and Monographs. 53. Providence, RI: American Mathematical Society (AMS). x, 618 p. (1997). ZBL0889.58001.]
A section $u$ of a bundle $(p,TM,M$ where $p: TM \longrightarrow M; \quad \text{$p$ is a smooth map})$ is a smooth mapping $u: M \longrightarrow TM$, with $p \circ u =\text{Id}_M$.
The support of the section $u$ is the closure of the set $\{x\in M \mid u(x) \neq 0_x\}$ in $M$. $\big(\text{$0_x$ is the null vector in $T_x(M), x\in M$}\big)$
Then the space of all such smooth sections of bundle $(p,TM,M)$ is denoted by $C^{\infty}(M \leftarrow TM)=\mathfrak{X}(M)=$ space of all smooth vector fields on $M$.
Again they define
The space of all such smooth sections of bundle $(p,TM,M)$ with compact support is denoted by $C_{\mathbf{c}}^{\infty}(M \leftarrow TM)=\mathfrak{X}_\mathbf{c}(M)=$ space of all smooth vector fields on $M$ with compact support.
Query:
What does compact support denotes here and what is the difference of $C_{\mathbf{c}}^{\infty}(M \leftarrow TM)=\mathfrak{X}_\mathbf{c}(M)$ with $C^{\infty}(M \leftarrow TM)=\mathfrak{X}(M)$?
Is it always necessary to have a compact manifold $M$ so that every element of $\mathfrak{X}(M)$ have compact support?