This is the time between the moment when you push the button and the image is actually captured. As such, it’s responsible for a lot of photos of fish butts. On modern compacts, this delay is usually because of the autofocus “hunting”, and it can get worse in low light. Autofocus can be set permanently “on”, but that can reduce battery life. This is also a factor in …

“Shot to Shot time”

… which is the time between taking one shot before the camera is able to take another. This may be very different if you are shooting RAW format, which can take 5-10 seconds to save. A good place to find stats on the performance of individual cameras is the dpreview website.

This used to be a constraint.
Once we got past about 5mp, however, it ceased to really be an issue, as these pictures will blow up to 12″x10″ without problem.

More detail can mean that you can “crop” the image more, but you’ll still probably be constrained by lens quality on a compact (let alone on a mobile phone !).

A high megapixel count can actually be a disadvantage if you’re shooting in RAW, as it reduces the number of images you can hold on a memory card.

You can check on DigiDeep

OK … there are a number of things to consider here.

Most digital compacts come with a zoom lens (ignore “digital zoom” - just look at the “optical zoom” range).
To let people compare like-with-like, there will normally be a “35mm equivalent” expressed.
A typical compact may have either a 35-105 (”3x”) or or a 28-115 (”4x”) zoom. Some now go to “18x” zooms.
These don’t necessarily give an advantage underwater, as particles in the water column generally make long zooming impractical.
It should also be borne in mind that the longer the zoom, the more complicated the lens, and this can affect the quality of the image, the “speed” of the lens, or both.

The “speed” of the lens is represented by the “Aperture Range”. Typically, this range may be displayed as “F2.8-F4.8″ (so … this would be “F2.8″ in wide-angle and “F4.8″ when zoomed in). Other things being equal, a low value is a good thing, as less light is absorbed by the lens.

Lens quality is one of the most difficult things to measure objectively.
Unlike the electronic components, it’s difficult for manufacturers to ensure consistency.
But it’s one of the single most important factors in the quality of your pictures.

If I had the choice, I’d choose a camera with AA batteries, rather than a proprietary battery.
Why ?
Well, generally they’re cheaper (and you’ll need a spare), and it means you only need to take one charger away with you.
This isn’t necessarily a deal-breaker on a camera.

First of all, it’s worth noting the difference between “Anti Blur” or “Anti shake” and true Image Stabilisation.
There’s a dpreview article which explains the difference well.
To summarise, most “Anti Blur” involves the camera using a shorter exposure time.
So this doesn’t reduce blurring where you may choose a long exposure.

Even this solution is compromised - the fast shutter speed is achieved by increasing the “ISO”. This will normally result in less detail. Again there’s a dpreview article which explains this.

None of these is necessarily bad - and it’s not a reason to exclude an otherwise good camera - but you should be aware that this is one area where the spec sheets can be misleading.