The blur is caused by the people moving while you were taking the photograph with a slow shutter. Honestly, I think it improves this particular photo a lot: it shows that the people are dancing, rather than just standing in weird positions.

If you want to, the only way to avoid it is to use a faster shutter speed. This necessarily involves compromises. If you use a faster shutter without changing any other settings, the photo will become dark. If you use a higher ISO, you'll get more noise. If you use a wider aperture (smaller f-number), you'll get less depth of field.

If you want to do a lot of party photography, it would probably be worth investing in a faster lens, i.e., one that allows you to use a wider aperture than the f/5.6 you were using. Fast zoom lenses can be very expensive – e.g., $1,500 for the Canon 16-35mm f/2.8 – so you might want to look at a fast prime lens. For example, a 28mm f/1.8 lens (Canon's is around $450) wide open would have let you shoot at about 1/100s without increasing ISO, but the depth of field might be a bit thin.

You should also seriously consider using flash. Good flashes aren't actually all that intrusive – as long as you don't use the camera's built-in flash, it doesn't need to do those annoying red-eye reduction flashes, and it shouldn't need to pre-flash for autofocusing. It's even less obtrusive if you're able to bounce the flash off the ceiling since, then, the bright bulb isn't pointed into your subjects' eyes. Further, basic use of a flash is straightforward and doesn't require much learning at all.