A differential form is closed iff its exterior derivative is $0$.
A differential $k$-form $w$ is exact iff there exists a differential $(k-1)$-form $\eta$ so that $\hbox{d}\eta = \omega$.
Every exact differential form is closed, but closed differential forms are only locally exact (Poincare's lemma).
The quotient of closed $k$-forms modulo exact $k$-forms is the $k$-th de Rham cohomology group $H_{\text{dR}}^k(M)$ for some manifold $M$.
What's an example of an obstruction that de Rham cohomology can help identify (and how does it identify it)?
(Possible examples: global integrability of $k$-forms, existence and integrability of vectors fields, existence of circle bundles, existence of fiber bundles, existence of global sections, existence of connections, etc.)
There are lots of examples, the one I've thought comes from symplectic geometry.
Let $M$ be a compact smooth manifold, a 2-form $\omega \in \Omega^2(M)$ is symplectic if it is non-degenerate (so if $\iota_X\omega$ is not the zero form unless $X = 0$) and closed. A symplectic manifold is a pair $(M,\omega)$.
The existence of such a non-degenerate form forces $M$ to be even-dimensional, say $\operatorname{dim}M = 2n$, one can check that non-degeneracy implies that $\omega^n$ is a volume form; in particular it can't be exact. Thus $[\omega]\neq 0$. Reading this reasoning backwards, we find that, if $H_{DR}^2(M) = 0$, then $M$ cannot be symplectic.