Project on Simple Groups I have to give a 15-20 minute presentation on a topic or result dealing with Simple Groups. 
Any ideas on what I can look into?
 A: Title: "Why do we care about simple groups?"
Your goal for this talk would be to convince the audience that understanding simple groups is equivalent to to understanding all finite groups. Why?
First, talk about Jordan-Holder. Note that all groups have a unique set of composition factors up to isomorphism, and that these factors are simple groups.
Second, talk about the Classification. Don't dwell too much on details. Say that there are 4 kinds of simple groups, the first two are easy and familiar: cyclic groups of prime order, alternating groups. Then mention that there are a bunch of families of groups lumped under the name "groups of Lie type," but don't elaborate. Finish with sporadic groups, emphasizing that classifying these took forever and were the difficult part of the Classification theorem, and we that only finished relatively recently.
The third part is the extension problem. Now that we know what the simple groups are, we know all the possible sets of composition factors a group can have.  If we knew how to determine a group up to isomorphism given its set of composition factors, we'd be done with finite group theory. Problem is, composition factors aren't unique- two groups can have the same set (take $\mathbb{Z}_2\times\mathbb{Z}_2$ and $\mathbb{Z}_4$, for example). So where do we go from here? The group extension problem is to answer, given groups $N$ and $M$, what all groups $G$ can occur with $M\unlhd G$ and $G/M\cong N$. If we understood these, then by induction and Jordan-Holder, we'd understand all finite groups.
So, that's why people like 'em.
A: Here are a couple of ideas, which could be discussed in tandem.
Composition Series and Extensions - Every finite group has a composition series.  The composition factors are simple and are independent of the chosen series.  This is the Jordan-Holder theorem.  Then, every group can be reconstructed by a series of extensions using its composition factors.  This allows us to view the finite simple groups as the basic building blocks of finite groups, just like prime numbers are the building blocks of the natural numbers.  The one main difference to highlight in this comparison is that "uniqueness of factorization" doesn't hold with finite groups.  For example, the composition factors of the two groups of order $4$ are identical.
Classification of Finite Simple Groups - Perhaps the crown jewel in the theory of finite groups.  The list of finite simple groups is straightforward to write down, but it is fairly surprising that the list is complete (why exactly 26 sporadic groups?).  The proofs of some deep theorems about finite groups are reduced to simple groups (using extensions and composition series), and then the classification allows one to check the result for all of the simple groups.  It would be appropriate to mention Burnside's Theorem and the Feit-Thompson Theorem here, as they are used heavily in the classification.
A: Besides the purely group theoretical backing as pointed out by Jared and Alexander Gruber, I would certainly mention Monstrous Moonshine, still a field of active research, where different fields (finite simple groups, modular forms, string theory) come together.
