I just posted about a comparison between a Point and Shoot and Medium format camera. After thinking about it, I realized that I have never defined what a point and shoot camera is and contrasted that with digital SLR. In the sections below, I will discuss these two types of cameras and the differences between them. I will provide some key considerations when choosing between them.
Point and Shoot
This is a type of camera that is an all in one solution and offers no lens interchangability. This is the most common type of camera out there. If you own a camera then the highest likelihood is that you own one of these. These cameras also tend to be highly automated and offer limited customization. That is, in most cameras, you cannot set things like aperture or shutter speed. This is not a problem per se since the average shooter does not care about these things, but can be limiting for more advanced shooters. They also use smaller sensors which provide weaker low light performance and often worse image quality than a Digital SLR
The other thing to remember is that since the lens is not interchangable, you are stuck with the lens that comes with the camera. You had better like it because there is no other option. Thus, you should consider whether the zoom range is enough for you.
I wrote more about P&S options in this post.
These cameras are most similar the ones you probably grew up with (or your parents grew up with.). It is a camera that you compose and focus by looking through a viewfinder. The will have one large lens on the front which you can remove and switch with a different one.
The benefit of these cameras is flexibility. The lens interchangability means that you can have multiple lenses and choose the best one for any given situation as compared to having one uniformly average. The other benefit is that these cameras have larger sensors and offer complete manual controls if the photographer wants to use them. All of the features in dSLRs can often be daunting for new photographers.
The biggest downside of dSLRs is there size. The cameras tend to be large and bulk. Add in a bag full of lenses and a flash and all of a sudden you have a potentially large kit to carry with you. Even the smallest dSLR and one lens will typically be much larger than one of the point and shoots described above.
If you ask me which you should purchase, the answer is that it depends on what you want. On one hand the point and shoots provide the most portability with a trade of low light performance and image quality. In contrast, the dSLR provides the best image quality with the trade off of size and complexity.
For most users, a point and shoot would be more than adequate. In fact an advanced amature may want to consider a camera like the Powershot G10 that offers full manual controls on a point and shoot form factor. A dSLR is a great choice for the photographer who really wants the ability to get the best image quality and ability to change camera settings.
Many photographers I know have both! They use the dSLR for those occasions where picture quality really matters and carrying the full kit is not a problem. They use a point and shoot for those occasions where they want to travel light and have a simplified photography experience