Actually, I use a professional digital SLR (without naming numbers, let's just say $$$$$ - I'm a photographer by trade).
At least one of the images above was taken with a $300 point-and-shoot Canon S230, however, so you can do a great deal with less expensive equipment.
I generally recommend cameras in the Canon line. You can find reviews in the DP review forums:http://www.dpreview.com
The Canon line (particularly the G-series, and for size, the S-series):http://www.dpreview.com/reviews/specs/Canon/
The main drawback of point-and-shoot cameras tend to be the small, harsh flash. To get the "best" light possible, you want a diffuse light source.
Bright (but not sunny) overcast days (the sky is like a big softbox)
Sun coming through a white curtain
Morning light (orange-tinted glow)
Bright white rooms
Cameras that use removeable flash units, which often can be tilted so the light is bounced off of the white ceiling.
Sunlight (if you have to)
Any type of flourescent (causes color cast)
Mixed flourescent and incandescent (small table lights)
Dark rooms where the flash is the sole source of light
The brighter a room is, the better the images will be. The less light there is, the more harsh shadows you'll have, because the tiny flash is made to do all of the work.
Also, think focus. Many cameras have a two-step focusing process where the user is supposed to depress the shutter release (button) half-way to focus the image and set the exposure, then a full press on the button takes the picture. To avoid camera shake and blurry pictures, steady your arm/elbow against your side whenever taking a picture. And realize that most point-and-shoot cameras weren't meant to take extreme closeups, so they probably won't be able to focus for very tight shots.
For example, the photo below:
I was in a car. I purposefully turned my flash OFF to avoid harsh light and harsh shadows. I wanted ambient, soft, natural light. A flash would have reflected off of her white fur. But then my shutter speed was too slow to take the picture without showing blurriness from the natural shake of my hand. The camera wants more light to properly expose the image, so it leaves the shutter open longer so more light can come through the lens. But then the film (or digital chip) records any movement as well. So I turned the flash off, but held my hand REALLY steady.
More light = higher shutter speed = sharper photo
Less light = needs flash to simulate sunlight = harsh shadows ***OR***
Less light + no flash = slower shutter speed = more camera shake = blurry photo
I hope that makes sense! (Everyone tends to have a different level of understanding of these things, so forgive me)