Many partitions of unity on sufficiently "nice"; what does this mean? In a class I am taking, we are told that are manifolds have "many" partitions of unity (we assume paracompact, Hausdorff, second-countable). However, the course content is not related to this subject. If I am trying to integrate a compactly supported function $f$ against a measure defined by a differential form and have a finite set of open sets $\{U_i\}$ sitting each in a coordinate patch, will there be partition unity so that I am able to do this?
Thank you for any help

The main reason to talk about paracompactness at all is that for any paracompact space partitions of unity exist. It is a rather elaborate work to actually construct them (using bump functions, amongst others), but once that result has been established you may freely use them in these casees.
It is surprising that a course where they are apparently often used doesn't actually show the existence of partitions of unity, while they're such an important tool at all. Since the construction is rather elaborate and there is good literature about it, I wouldn't like doing that here myself. I suggest reading the seventh chapter of the book "Introduction to differential topology" by Bröcker and Jänich, which deals with the existence of partitions of unity. There should be many other good sources out there, but I like this book in particular.
After you've done that: just think of paracompact spaces as "spaces having partitions of unity".
