all sorts will fit your need. but you may want one that's really small and portable, hence easy to fit in your pocket at parties and things they may go to.
Fuji Z1 is a great little camera, as is the F10, also by Fuji.
lots of the casios are also very good. I have the Z750, which is brilliant. but may offer more than you really need.
I'd advise you to go into a Jessops store (or some good independant one), and talk to someone who knows what they're talking about, and so that you can have a mess about with the cameras and see which ones feel good to you.
In terms of quality, a 3-4 megapixel camera will do you fine. But the more the better generally. And go for a 3x optical zoom, again the more the better, but that should do fine. Almost all of them will have a feature for a fast shutter, for taking pictures of kids running about. Also worth seeing how fast the cameras start up. Some are near instant (like mine), others take a second or so to turn on and actually be ready to take a picture.