Compared to point-and-shoot (POS) cameras, dSLRs have a few drawbacks and LOTS of benefits. A dSLR won't guarantee you better pictures, but it will give you a lot more options and more control if you're willing to put the time into it.
First the drawbacks (of dSLRs):
1) dSLRs are usually more expensive.
2) dSLRs are usually bigger and heavier. This is certainly a drawback when your arm gets tired, but it may sometimes be a benefit since it may be easier to hold a heavier camera steady in some circumstances.
Now the benefits (of dSLRs):
1) Changeable lenses. This is certainly the most obvious benefit, but it tends to obscure the fact that there are other benefits - even if you never buy another lens.
2) Optical viewfinder. Most dSLRs have one while most POS cameras do not. Personally I don't use it anyway, but some people can't live without them.
3) Manual focus override. Many (but not all) dSLR lenses will let you focus manually if you want to - which is very useful when the autofocus gets it wrong. This happens especially often in low light. Some even let you manually adjust the focus AFTER the autofocus has it's turn (both the camera and the specific lens must offer this option - many do). POS cameras never seem to offer this option.
4) Better autofocus in low light. Most dSLRs use a different focus mechanism than POS cameras. The result is that most dSLRs focus faster and much more reliably in low light.
5) Better sensor. Most dSLRs have a much larger sensor than most POS cameras. (We're not talking megapixels but the actual physical size of the sensor). The result is that, in low light, a dSLR will usually take a better picture with less visible noise than a POS camera at similar settings. This tends to not be noticeable in bright light and to be VERY obvious at higher ISO settings (400 and above). The difference is very obvious, especially if you take lots of indoor shots without a flash. So, AT THE SAME SETTINGS, a dSLR will give you a much better picture.
6) Better lenses. Of course, a dSLR lets you buy better lenses than the one that came with your POS camera, but even the "cheap kit lens" that came with it is probably very good when compared to most POS cameras.
7) Manual controls. Some POS cameras give you some manual control, others give you virtually none, while most dSLRs give you a LOT of manual control. Today, most of them also do as much automatic stuff as a POS, but it's nice to have the options for manual control when you're ready to try them.
Battery life. Some dSLRs give you very good battery life. My Nikon d90 will let you take 700-800 (or even more) non flash pictures on a single charge. Most POS cameras don't give you nearly that many shots.
9) Start up. This is the time from when you hit the ON switch to when you get the first picture. Most dSLRs are much faster than most POS cameras in this regard.
10) Burst rate. This is how fast you can take several pictures in a row by holding the shutter down (like a paparazzo). This really matters for sports shots and animal shots (among other things). Cameras vary, but most POS cameras are VERY slow to take multiple sequential pictures, while dSLRs usually range from pretty fast to downright scary.
11) External flash. Almost all dSLRs have a hot shoe on top for an external flash. Almost all POS cameras do NOT. This is definitely something you might want to move up to later.
As a new photographer, I would suggest buying a low-range dSLR to start with ($500-$700), with a single "kit zoom lens". Quite frankly, once you get used to the features and flexibility, you're probably going to know enough to have an opinion about what features you care about, and which one you REALLY want. Therefore, you should consider your camera to be a "starter", and not spend too much money on it or accessories until you've practiced for a while and know more about your specific needs and wants. I would recommend a lower model from a major vendor like Nikon or Canon (since there is a learning curve, you'll save time and learning effort by moving "upline" with the same vendor, and be able to keep at least some of the accessories and lenses you buy for your current one). It will also make a good spare once you buy your "good" camera, or you can pass it on to the family Note than lenses are generally NOT interchangeable between vendors, and neither are the new "smart" flash units, so assume they are not unless you know otherwise.