Unfortunately, resolution is seemingly one of those simple-to-understand "more is better" factors. So camera manufacturers have competed on the basis of resolution, way beyond the useful point for most people. Especially in compact cameras with small sensors, the resulting pixel sizes are so small as to compromise quality. So the bottom line is that any camera these days will have more than enough resolution.
Putting that aside, it seems like 2 megapixels is the minimum useful resolution for most people. More is better, but it may not matter much if you're mainly going to be putting pictures on the web, or getting the occasional 4x6 printed. I do recommend taking pictures at full resolution and lowest compression if you can afford the space. After all, if you don't store those pixels at first, there's no way of getting them back!
Note that cropping throws away pixels, so if you're going to do drastic cropping, which is basically the equivalent of high-power digital zooming, you may need more pixels up-front to end up with enough for even high-quality web images.
Even just a few years ago, anything more than 10 megapixels was seriously high-end, and reckoned to match or even exceed effective 35mm film resolution. These days 12+ megapixels is pretty normal for even medium-range cameras, and the high-end is 20+ megapixels.
The problem with higher resolution in smaller cameras is that smaller pixels gather less light apiece, and so require more amplification and are thus electrically noisier. Digital SLRs have larger sensors which mitigates this phenomenon, and yield much smoother results, especially at ASA 400 speed and above.
In a word, Yes.
Many cameras have a so-called "macro" mode. You may not need this for most situations, since the "standard" focusing distance may be as close as 2 feet, and you wouldn't want to go below that for portraits of people, say.
But if you do want to take a picture of the occasional flower or fine detail from just a few inches away, then macro or close-up is useful.
Like any other optional mode, though, one has to be careful not to leave the camera in macro mode for normal shooting. Everything will come out blurry. Ideally, one would notice this quickly by reviewing the LCD screen, but it may not be that obvious in a small screen.
I had previously dismissed "movie mode" as a gimmick, or just not my bag, but I'm warming up to it a little. The first thing to be said is that this capability, becoming more universal, does not by any means turn your camera into a real digital video camera: you are typically limited to short clips (say 30 seconds) of low-resolution (say 240-by-320 pixels) video. This can be fun for a "novelty" reminder of very dynamic events, but if you're really interested in video, you'll need to buy a dedicated video camera (which in turn will take mediocre stills; such is life).
(As I write in 2008, the situation is changing: still cameras are getting more capable, being capable of higher-resolution and longer videos. They still typically have limitations relative to real video cameras, though, such as often not adjusting focus while taking a video.)
Perhaps more useful is a "motor drive" mode that lets you take a number of full-size still images in rapid succession. Because of internal memory buffer constraints, there may be a pretty small limit on the length of such a "burst". (Memory cards are actually fairly slow; it can take seconds to write an image to the card, or retrieve it for display/download.) Again, this is where more expensive cameras do better: they have larger internal buffers and more sophisticated electronics to process images faster and interface with faster memory cards.
I suspect that most people will just leave their camera on full-auto all the time. If you have special needs, like long exposures, or non-standard flash synchronization, then you probably don't need my help :-)
One exception to this is probably control over the flash. There are situations in which you'll want to disable the flash, and take your chances with a longer exposure. Or you may not want the distraction of multi-flash red-eye reduction when you're not taking pictures of people. Most or all cameras will support this kind of control; the real issue is how convenient it is: just a button press, or navigation through a complex on-screen menu.
Short of full manual control, it may also be useful to have more control of exposure and focusing. Examples would be backlight compensation, spot-metering, and metering and focusing "hold" options, which allow one to select some small or non-central element as the deciding point. And even fully-automatic focus and exposure systems differ in their sophistication and effectivness, with the best being "matrix" or "evaluative" systems that use a larger number of decision points, not just the center, or an average over the image.
As mentioned previously, digital sensors by default are fairly "slow", so it can be useful to be able to select a higher ASA equivalent speed. But like using faster film, "grain" or image noise increases and smooth tonality suffers.
"Playback" modes are something unique to digital cameras. It is useful to get a quick view of the picture just taken, so you can judge the effect of flash, the subjects having moved, etc. Most cameras also allow you to zoom in and navigate to check finer detail, delete pictures you don't want, etc. Most these days will also allow you to print directly from the camera, setting various attributes of the resulting print. I've never used the latter.
In my opinion, digital zoom is worthless for taking pictures, since one can achieve the same effect in a more general way by cropping and re-sizing in any image manipulation program. Optical zoom is definitely valuable. For example, portraits are generally more pleasing if they're taken with a longer-than-standard focal length, from further away, rather than closer with a wide angle. (The latter tends to make noses look bigger!)
For comparison, with a full-frame 35mm camera, a "standard" lens is 50mm, the "classic" portrait lens is anywhere from 85mm to 105mm, telephotos range up to 200mm or 300mm (with exotic and expensive lenses going beyond this), and wide-angles range from 35mm down to 20mm or even 14mm (exotic/expensive).
The minumum tends to be a 2x optical ("real") zoom, from 35-70mm (not really, but those are the equivalent focal lengths for a 35mm camera). That's a mild wide-angle to a mild telephoto, and I would always rack it out to full telephoto for a portrait of a single person. (Cheaper or speciality cameras may have single- of fixed-focal-length lenses.)
For a larger group, with the people 6 feet away or more, then wide-angle is fine. So if you are going to do individual portraits, a fixed focal length, which is typically somewhat wide-angle, may prove unsatisfactory. And of course for a wide range of subjects, from buildings to landscape to whatever, one sometimes wants to be more wide, or more tight.
More typical these days is 3x or 4x. These tend not to be that wide on the wide end, 35mm or if you're lucky 28mm equivalent, and offer the extra extension on the telephoto end. For stationary subjects one can use panorama software to get wider (and this is the only alternative for really wide situations, even 360 degrees!). And one can crop to get tighter, but that loses quality. The downside to a wider optical zoom range, up to 10x in some cameras, is size, weight, expense, possibly lower optical quality, and especially on the "long" or telephoto end, a very "slow" lens that requires a brightly-lit situation to work well. (The latter issue is exacerbated by the need for faster shutter speeds, or flash, to avoid camera shake with long lenses, since they magnify more.)
Bottom-line, for most people 2x (optical) zoom is the minimum they should consider, 3x will probably be fine, 4x is nice especially if it starts fairly wide (28mm equivalent), and one should consider the tradeoffs before going for something exotic like 10x. Some cameras can take optional adapters to make the lens have a wider or longer focal length, but those are rather clumsy, and expensive.
Lens speed is important, especially since digital sensors are pretty slow by default. The fastest lenses on consumer cameras are around f/1.8 or f/2.0 at the widest setting, which is good. A maximum aperture of f/2.8 is OK, but a maximum aperture much below f/4.0 gets problematic in lower-light situations. And note that the maximum aperture is at the wide-angle end of the zoom range; the telephoto end will in general have a smaller (numerically higher) aperture (slower) which unfortunately is the opposite from what one would want.
Copyright Richard Schooler, 2001-2008 schooler@alum.mit.edu