John,
 If you use ONLY Aperture Priority Mode then welcome to the world of 
blurry pictures.
Explanation:
 Most Poit and shoot digital cameras have pretty short focal length 
lenses. This means that there is a very good depth of field (when the 
zoom is less that 50mm 35mm film camera equivalent). Using aperture 
priority increases the aperture and gives you an even greater depth of 
field.
This has to come from somewhere as the aperture, shutter speed and ISO 
Film speed equivalent have a fixed relationship. (See below)
If  x = f8 @ 1/125 @ 200iso
then
   x = f4  @ 1/500 @ 200iso
Where x = the amount of light falling on the film/ccd.
         1/125, 1/500 = shutter speed
         f8, f4 = Aperture
         200iso = equivalent to a film speed of ISO 200.
As the speed goes up(ie shorter exposure), the aperture goes down.(less 
depth of field)
As the Aperture increases  (more depth of field) the shutter speed slows 
down = more chance of blurry images if there is any movement in the frame.
As a general rule of thumb, 1/250 is fast enough to freeze most movement 
in People.
However, many point and shoot cameras have very little control of what 
aperture is used in AP mode.
In general and domestic photography, it might be better to go with 
Shutter Priority. Do a few tests of each Priority in the same situation.
The main tip I can pass on here to all 'snappers' is to know the 
limitations of your camera.
I spend quite a lot of time 'playing around' with any new piece of kit I 
buy before I use it in anger.
I found this out when armed with a brand new Praktika LTL 35mm Film 
camera I went to a concert at the Brighton Dome on 20th Jan 1972. Most 
of the pictures were well underexposed. However the 4 frames that I took 
when Roger Waters did the scream during 'Careful with that Axe Eugene' 
are some of the pictures I am to this day the most proud of.  As a 
slight aside, this was the first ever public performance of 'Dark Side 
of the Moon'. This was recoded by the Stones Mobile Studio which was 
parked outside.
Tickets cost 2.50gbp.
I did later forget this lesson when I went to Madagascar in 2004. My 
then brand new Nikon D70 was overexposing by 1.5stops. Using it to 
photograph the white bodies of Red Tailed Tropic birds was a complete 
fiasco.
Stephen D