<p>What is an NP-complete problem? Why is it such an important topic in computer science?</p>

Honestly,
Wikipedia
might be the best place to look for an answer to this.
If NP = P, then we can solve very hard problems much faster than we thought we could before.  If we solve only one NP-Complete problem in P (polynomial) time, then it can be applied to all other problems in the NP-Complete category.