Digital SLR vs "something else"
I'm not sure that the alternative to a digital SLR is a compact camera, so I'll address the question from a different perspective. In fact, the alternative to a DSLR is usually considered to be what the camera makers call a "Super Zoom" camera and not a compact camera.
There are really three advantages of a DSLR. The first two are inherint to the DSLR design, the third is coincidental (e.g. a non DSLR COULD have this same advantage, but none do at this time that I am aware of).
The first advantage is interchangeable lenses. If you want that, you NEED a DSLR. Period. But the fact is, most people DO NOT want that, although true professional photographers do. For myself, I want a good quality, high ratio zoom (at least 10:1; roughly at least 30mm to 300mm (these are 35mm equivalents and not necessarily actual focal lengths). The truth is that the quality of any such lens will not be as good as two or three separate, interchangeable lenses covering that range, which is why a true professional would select a DSLR. In fact, it is incredibly difficult to make just a good to very good lens like that, but a few firms do it (Tamron probably makes the best such lenses for DSLRs, but they are not cheap ... $400 to $600 just for the lens (the camera is additional), and even at that, again, they are not as good as several separate lenes covering the same range). However, for even a serious amateur, this is probalby a better solution than multiple lenses. I mean, when you are on vacation with your family, do you want to be a photographer of the vacation (as you futz with a 20 pound camera bag and changing lenses) or do you want to be a participant in the vacation who happens to be taking some pictures?
The second advantage of a DSLR camera is that it has an optical viewfinder instead of electronic viewfinder. To be honest, the fact that people consider this to be important seems to me to be an indication of their lack of understanding of what is going on inside the camera. In my view, an electronic viewfinder SHOULD be PREFERABLE, for a number of reasons. But the fact is that the electronic viewfinder is an area where the camera makers cut cost, so SOME (ok, perhaps many, perhaps even most) electronic viewfinders have low resolution and low refresh rates. And on COMPACT cameras, there isn't even an electronic viewfinder, in most cases, there is only an LCD screen, which may be useless in outdoor sunlight, and objectionably bright (particularly to other people) in a dark theatre. But most "super zooms" have both an LCD screen and an electronic viewfinder, and SOME of the electronic viewfinders are even pretty good.
The third advantage of DSLRs, which is NOT inherint to the DSLR type camera (but I don't know of a non-DSLR currently in production that offers this) is a very large image sensor: on the best DSLRs, an Image sensor the size of a full 35mm film frame, or, on many others, one slightly smaller, but still approximately the size of APS film frame. Compare that to the typical non DSLR camera, whose image sensor is the size of the eraser on a #2 lead pencil (but not round, obviously). For a variety of reasons, the bigger the image sensor, the better the picture quality, especially as regards low light levels and ability to give you high sensitivity (high ISO equivalent ratings) with low noise levels.
Note, however, that the larger the image sensor, the more difficult it is to make a good qualtiy, fast, high-ratio zoom. Again, Tamron seems to be best at this, because they started making 10:1 zooms back in the 1990 for SLR ***FILM*** cameras, and they know how to do it about as well as anyone. BUT, again, these are expensive lenses which are still not as good as multiple lenses covering the same range.
You mentioned as the alternative to a DSLR a "compact" camera, but there is another choice, as I mentioned, a "Super Zoom", which is a MUCH larger non DSLR camera that looks like a DSLR, has most of the features of a DSLR and is the size, weight and complexity (in manual or semi-automatic mode) of a DSLR but it's not an SLR and it has a single fixed high-ratio zoom lens. But the downside of this is that these are still using "small" image sensors, and, consequently, they don't have the quality of a true DSLR.
The downsides to a DSLR are this:
1. Cost (camera AND lens)
2. Opportunity for contamination of the sensor, which is exposed while changing lenses
3. MANY DSLRs can't shoot video (but some can)
4. Size, weight and complexity (compared to a compact camera, but not necessarily compared to a "super zoom")
This is a very personal decision, and I'm not going to recommend which way is right for you. But I hope that this has given you some background information with which to make a better decision.