This is a question I hear at least once per wedding from someone's date or Uncle Bob. Here's my answer.
Canon or Nikon?
It doesn't matter.
That's a somewhat controversial opinion, but as far as I can tell Canon and Nikon are like Coke and Pepsi. Different, but basically interchangeable. I'll recommend a Canon because that's what I use, a Nikon user will suggest Nikon, and both will be perfectly good.
Whichever you pick, you'll stick with forever, because you'll have bought Canon lenses, flashes, etc. and it would cost too much to switch.
Ok, so what should I buy?
Get the newest Canon Rebel. It's their entry-level camera. It was my first DSLR, and I used it for five years, including the start of my professional career. It's a great camera, light, easy to use but has all the features you'd want in a DSLR. It's an affordable way to get started.
My old Rebel, about a decade old now, was my first "real" digital camera. Still works.
Then, if you feel like you haven't spent enough, invest in better lenses. You'll get the most bang for your buck from prime lenses (ones which don't zoom). You can get very affordable primes with good quality glass and wide apertures.
Apertures?
The aperture is the hole that lets light in through your lens. It's described by the f-stop - the lower the number, the wider the hole. Low f-stop lenses are great - they let you shoot in darker situations, and give you the wonderful, blurry backgrounds that everyone loves.
The Canon 85mm f/1.8 prime is a great lens for portraits, for example.
And a Rebel will get me great photos?
No.
I mean, sure, they'll be better than your cell phone. But a better camera doesn't make you a great photographer. Practice does.
Which is why I'd suggest going a different route. Get the newest or second-newest Canon S-Series.
It's a point and shoot, small enough to fit in a pocket, which means it fits the old adage, "The best camera is the one you have with you." I carry my S110 with me every day.
It's not just portable, either. The s-series is a powerful little camera, with tons of manual controls. it also shoots in RAW, the highest quality file format.
You can learn almost everything you would on a Rebel from the s-series, but it fits in your pocket, and it's cheap enough that if it gets lost, stolen, or broken (something I know nothing about...), it's not the end of the world.
By far the best bang for your average photographers buck.
(Note: The links to specific recommendations are up-to-date as of the posting of this entry, but may be out of date in the future.)