A) For a differential manifold $X$ the following are equivalent:
a) X is paracompact
b) X has differentiable partitions of unity
c) X is metrizable
d) Each connected component of X is second countable
e) Each connected component of X is $\sigma$-compact
Partitions of unity are a fundamental tool in all of differential geometry (cf. kahen's answer) and would suffice to justify these conditions but the other equivalent properties can also be quite useful .
B) However occasionally non paracompact manifolds have been studied too. For example:
1) In dimension $1$ you have the long line obtained roughly by taking the first uncountable ordinal set and adding open segments $(0,1)$ between its successive points.
2) In dimension $2$ there exist non paracompact differentiable surface ( Prüfer and Radò). However every Riemann surface, that is a holomorphic manifold of complex dimension $1$ and thus real dimension $2$, is automatically paracompact.
3) Calabi and Rosenlicht have introduced a complex manifold of complex dimension $2$ which is not paracompact .
Edit As an answer to Daniel's question in the comments below, here are a few random examples of consequences of the existence of partitions of unity on a differential manifold $M$ of dimension $n$.
$\bullet$ If $M$ is orientable it has an everywhere non-vanishing differential form $\omega\in \Omega^n(M)$ of degree $n$.
$\bullet$ If $M$ is oriented you can define the integral $\int_M\eta$ of any compactly supported differential form $\eta\in \Omega^n_c(M)$ of degree $n$.
$\bullet$ The manifold $M$ can be endowed with a Riemannian metric.
$\bullet$ Every vector bundle on $M$ is isomorphic to its dual bundle.
$\bullet$ Every subbundle of a vector bundle on $M$ is a direct summand.
A sophisticated point of view (very optional !)
All sheaves of $C^\infty_M$-modules (for example locally free ones, which correspond to vector bundles) are acyclic in the presence of partitions of unity.
This has as a consequence that paracompact manifolds behave like affine algebraic varieties or Stein manifolds in that you can apply to them the analogue of Cartan-Serre's theorems A and B.
This is, in my opinion, the deep reason for the usefulness of partitions of unity on a manifold. (The last bullet for example was directly inspired from its analogue on affine varieties or Stein manifolds)