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Lens quality basically will dictate the image quality. This is something that is very specific to each camera.
General build quality will also ensure a long lifetime.
Plus you want few handy functions. Personally an optical viewfinder is very helpful.
That one costs slightly more. The reason I suggest these two is because canon compact point and shoots are the best there are in that class, and these two stand out from there other models because of their 3 inch lcds and fairly good optical zoom.

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Compact Point and shoots: These are your small credit card sized cameras that most people think of these days when they think of cameras, available at any electronic store pretty much. Very portable and stylish.
This is one of the best point and shoot cameras out so far and costs around $250 I believe. The other good one, which is like an upgrade of the SD 880 is the canon SD 970 IS. Since most people who use point and shoot cameras don't use the view finder, these models got rid of that and put a huge and vibrant 3” LCD instead, a smart move.

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Also they have tons of features that are more than any casual photographer needs. Check them out on the canon website, then buy from reputable dealers.
Fujifilm is alright, however suffers from purple fringing a lot on many of their cameras.
I would say these are the way to go if you are a casual photographer and just take pictures when you are out with friends, or go for trips, etc. In the compact point and shoot range I would suggest Canon SD 880 IS (comes in gold and sliver).

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We have reached the point for which most people have enough megapixels, and to be honest the megapixel count means jack. Two things you need to look at are both quality related.
And I personally absolutely must have full manual functions including manual focus and aperture control. Plus a hot shoe and thread for filters are essential.