I don't think anybody mentioned off-camera flash yet. On-camera flash, even with a diffuser or a softbox, illuminates straight ahead, and although you can make the light softer, it still has a very cliche look in my opinion. I almost always use a flash with a diffuser off-camera using the OC-E3 TTL cord in a party situation. I use just a simple sto-fen omnibounce (lately I've been messing around with a LightSphere that a friend gave me), and hold it in my left hand as I shoot. It's a bit more cumbersome, as both of your hands will be full, but I think the results are worth it, just because they look different from most other head-on flash party shots.
Gelled flash is another great technique that takes some experimentation. Again, even if you get the lighting down and are able to capture good moments, some environments just look boring, and I sometimes use gelled flash to give a color tone to the background. In a nutshell, you have to think about what color you want the background to be tinted, and then gel your flash with the "opposite" color. If you want a green background - go for magenta gel, if you want it to be reddish, use a green gel. Then in post, you can use white balance and other tools to bring your flash-lit subject back to "neutral" light (and it's up to you how natural you want them to look), as the background gets colored. (You could also use manual WB in-camera, but I find that these photos almost always need color adjustment in post anyway). If you are really pushing the color, Lightroom might not be enough - its WB sliders will max out. Photoshop and LAB mode are your friends for this approach. It can be laborious, but again, the result will be a more unique look for even a generic space:
Agreed with others to bump up the ISO, open the aperture, and slow down the shutter to get maximum ambiance - then dial them back as needed.
First two photos are just bounced flash in my left hand. The second two illustrate the gel method.