Point and shoot and digital SLR cameras are worlds apart in more ways than one. The first question you have to ask yourself is what kind of photography are you going to be doing? Do you want something where you can just press a button and forget about it, or do you want to jump into your shots head first with complete control over every aspect? There are upsides and downsides of both dSLRs and point and shoots, which will be outlined here. In general, however, point and shoots offer more portability and less interaction, while digital SLRs offer better quality and more control. An 8 megapixel dSLR will likely be much better quality wise than a 10 megapixel point and shoot. This is due to the fact that dSLRs have larger CCD sensors, and much more advanced optics. However, you can buy the best digital SLR in the world, but it is only going to be as good as the quality of lens that you put on it. It's also worth noting that if you are in the market for a digital SLR, you may want to look at one of the (slightly) lower priced brands such as Olympus or Sony, as they usually have slightly more features than the entry level Canon and Nikon SLRs. However; accessories can be slightly harder to find for these brands of SLRs.
Pros and Cons
Point and shoots and digital SLRs are tailored towards two completely different types of photographers. With the pros and cons outlined below, you may be able to find out which would suit you better.
One of the most important factors for many people when purchasing a camera is the cost. However, not many people think about cost of repairs, upkeep, and accessories down the road... maybe you should.
Point and Shoot: Lower initial cost, and lower cost for repairs.
Digital SLR: High initial cost, repairs will be more expensive, and accessories down the road (such as lenses and filters) will drain your wallet.
This one is somewhat tricky. Point and shoots vary extremely in quality, while most dSLRs are pretty good quality.
Point and Shoot: Lower image quality, many have plastic lenses, smaller sensor is more susceptible to grainy pictures.
Digital SLR: High quality internals, high quality optics, large sensor is less susceptible to grain.
Ease of Use
Point and shoots are really and truly the only camera for beginners who are not interested in taking the time to learn the ins and outs of photography.
Point and Shoot: Extremely easy to use and beginner friendly.
Digital SLR: Steep learning curve, but once you get it down, it's cake. Higher interaction level may frustrate some people.
Nothing holds a candle to digital SLRs in this category. Point and shoots can't even come close.
Point and Shoot: Allows some customization of camera settings, but it stops there.
Digital SLR: Complete control over every aspect of your shot, tons of accessories to further customize your shots, continuous shooting works great for action shots.
Again, the point and shoot can't even compete in this category. Digital SLRs have a plethora of available accessories.
Point and Shoot: Very small accessory market.
Digital SLR: Extreme amount of accessories available. Lenses, lens filters, external flashes, flash diffusers, lens hoods... I could go on forever.
Both are really tied here, there's no major reliability issues with either.
Point and Shoot: Reliable as long as you get a good quality brand.
Digital SLR: Reliable in general, but higher cost to fix if something does happen to break.
Maintenance and Upkeep
The digital SLR definitely requires more interaction in this area, but nothing extreme at all.
Point and Shoot: Next to no upkeep needed. Clean the lens every once in a while and you're good to go.
Digital SLR: Larger lens surface area means it needs to be cleaned more often, sensor also has to be cleaned every few months (but this only takes literally a minute of your time, and a lens bulb to blow dust off).