Time Lapse Photography by Camera
Think of the typical time lapse photo, those long fluid lines of water rippling down a stream... The faster the movement that you are attempting to take time lapse photography of, the easier it will be for your point & shoot. While some time lapse effects may remain unattainable without more invasive means, like those wonderfully smeared clouds, some are still within your grasp without much effort.
Depending on the point & shoot, you may or may not have some control over the exposure length. If you are at all uncertain, please read the manual, as unlocking such precision controls is imperative.
Whatever control you do have over your settings, take advantage of them. Make the exposure time as long as possible, while to compensate make settings like ISO and f-stop lighten the image as much as possible so the image isn't overly dark. Make sure the flash is off!
If you can't set exposure or otherwise don't have manual mode, try pointing the camera at a dark place so that the auto mode will set a longer exposure time, and then doing the shot itself at whatever you're trying to expose for. This will work optimally under twilight conditions, where the low lighting will be conducive to longer shutter speeds (seeing that this often involves beautiful sunset and sunrise colors, you're hardly put at any disadvantage!) Also, again, the faster the movement is, the better this will work. Waterfalls can turn out wonderfully at even a ¼ second exposure! This is a useful trick for manipulating non-manual cameras in general.
If you want multiple short exposures instead, without extremely invasive hacks you're pretty much out of luck—unless you've got a timer and a lot of patience.
For general tips on composing time lapse photography images, check out this article.