Something to keep in mind is that when you buy a point & shoot camera, you're buying a "camera". It's not a modular system -- so whatever it does is all that it'll ever be able to do.
That's not the same when you move to camera with removable lenses. You should think of these less as "buying a camera" and more as "buying a camera system". So an important factor isn't what the camera does ... but what the system is capable of doing.
Canon & Nikon are the 800 lb gorillas of the space. They have a HUGE selection of lenses (more than you'll ever buy) as well as other accessories -- lenses, battery grips, etc. etc. They also enjoy very good third party support (everybody else makes stuff that works with these cameras.)
Other vendors make good gear, but the overall systems are a bit more limited. For this reason I typically don't recommend Olympus, Pentax, or Sony (Sony probably makes the most of the three.)
Assuming you'll look at Canon & Nikon, and assuming you want a DSLR (you didn't explicitly state that so this is a bit of a leap), they both group cameras into "entry level", "mid-level", and "pro" categories. If it's from Canon and it has the word "Rebel" in the name, it's an entry level body (the latest & greatest is the T4i). From Nikon, the 4 digit bodies starting with 3 and 5 are the entry level bodies (e.g. 3000, 3100, 3200, 5000, & 5100). The 3xxx are lower in the entry level and the 5xxx are higher in the entry level *except* they did just introduce the D3200 and have not yet countered with anything above the D5100, and the D3200 is pretty nice.)
Lastly: Lots of shoppers get into this paralysis of thinking that there is a "best" camera out there and they want it... and any other camera will be a disastrous mistake. That's just patently untrue. No matter what you buy, there's always a higher end model (there are cameras out there that cost more than your car.) All of the cameras cover the basics quite well. Of the DSLR bodies currently sold (not counting discontinued models), all of them will wipe the floor with whatever point & shoot camera you might have had in the past. Basically... prepare to be quite pleased with whatever you buy.
I shoot Canon, but I'd be equally happy with a Nikon. Neither brand is the "best". If it were, then when you look at the pro photographers out there you'd be seeing an obvious bias of one brand being used substantially more than the others. You really don't see a brand bias when you survey what they have... although there is an obvious bias that they tend to shoot using "pro" level bodies, you'll largely find huge numbers shooting either Canon or Nikon... and then a small number of them shooting with something else.
There is a myriad of different cameras out there.
