"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.