One of the situations where compact cameras really struggle is low light situations. The following are some examples of what happens when you shoot at high ISOs in (usually) in low light. See the grainy, ugly pattern over the whole picture? (Click on the picture to see it larger.) That is noise caused by pushing a small sensor too far. The higher the ISO value, the less light the sensor has to work with. Have a look at the ISO article for a more detailed explanation.
Fortunately, there is a solution to this noise problem. Buy a larger sensor camera, such as a DSLR where the larger sensor size gives much much better low light performance. Take a look below for similar shots but with a Nikon D40 instead of a compact camera, at an even higher ISO. Even though the D40 is now quite out of date, the low light performance is still light years ahead of the current compacts, as you can compare above and below.
Want this image quality but don’t want to carry a DSLR? There are now some interesting options coming onto the market, in the form of the “mirrorless” cameras. In essence, these are physically much smaller than DSLR’s while having the same sized sensors. You can check out the Panasonic G1, GF-1, G2, G-10, Olympus E-P1, E-P2, E-PL1 and Sony NEX-3, NEX-5. Even though they are still not what you could call pocketable, they are definitely smaller than a DSLR. This is a relatively new market, so the prices are relatively high for a cut-down DSLR, but should drop towards the end of the year as more players enter this market.