When I go to conferences and try to network, I want to be able to explain my research in a clear and concise manner while still effectively communicating what I do. What are the most effective ways of doing this?
An effective elevator pitch is between 30 and 60 seconds in length. Based on how most people speak, that's between 60 and 120 words. That's not a lot of time or verbiage, as this corresponds to about one paragraph in length.
If you look at the challenge as answering the six fundamental questions (Who?, What? Where?, When?, Why?, and How?), several of them have already obvious answers (these would be Who?—You!, When?—Now!, and Where?—My PhD project!). That leaves the three remaining questions: What?, Why?, and How?
Basically, I'd format the pitch as follows:
The other thing is to rehearse it until you can tell it without thinking about it—either because you have a rehearsed script, or because the key points have become ingrained and recallable without much thought.
POSTSCRIPT: As Artem mentions in a comment below, you should have different elevator pitches for different audiences. At a minimum, you should have one for experts in your specific field, one for other members of your profession but outside your field, and then a third for "general" audiences. Having one for family and friends who want to know "what exactly is is that you do?" probably isn't a bad idea, either.