Motivation of paracompactness

"A paracompact space is a topological space in which every open cover admits a locally finite open refinement" is the definition of paracompactness on Wikipedia.

Comparing with the definition of compactness, "a topological space is called compact if each of its open covers has a finite subcover".

In a first understanding, the difference you notice is that, to be compact, the space must have a finite subcover for EACH of its open covers, so every compact space would be paracompact.

I would like to know what is the motivation on that definition, in what way that concept helps in "extending" the notion of compactness and some examples in which paracompactness is an important property; if they are examples from differential topology or functional analysis, even better. Thanks in advance.

As Mike Miller said, paracompactness gives you partitions of unity, and anything you might want to do uses partitions of unity. Let me just give a short list of desirable properties a paracompact smooth manifold enjoys.

One can always define a Riemannian metric, and hence an ordinary distance function (non-paracompact spaces are not metrizable!)

Every real vector bundle is isomorphic to its dual (this is essentially the point above).

Definition of integration of compactly supported forms on an oriented manifold requires paracompactness.

Vector bundles over a manifold can be classified: A vector bundle over the manifold corresponds to the pullback of the tautological bundle over some grassmannian by some map. This does not hold for non-paracompact manifolds.