What is a single lens reflex digital camera?

As the other (as of this time) responder stated, it is a single lens that when you are looking thru the viewfinder, you are looking thru the lens, not a simulated or digital representation of what you should get, you see exactly what you will get.

SLRs have many benefits, and a few negatives, on the plus side:
They tend to be much faster cameras, my Canon 20d can shoot up to 5 shots a second.
You get a ton more control on the creative parts of the camera, aperature, focusing, ect are all at your fingertips
They are customizable, you can swap lenses to suit the type of shot you want to take, you want a long range shot of the squirrel on the tree across the park? pull out your telephoto zoom image stabalized lens and start shooting, you are going to take photos of people posing, pull out the 50 prime, and get to work.
SLRs have tons of gadgets you can add to them to do exactly what you need, bigger flash, external remote? all possible.
DSLRs (DigitalSLRs) usually use compact flash memory cards, on average, those are going to be the bigger storage capacities, I paid 150 last year for a 4 gig microdrive for my camera,and I love it. I get 1000+ 8 megapixel shots before it fills up.
DSLRs start at a minimum of 6.3 megapixel nowadays, and have much larger lenses to get more light in and better focus on what you are shooting.
Changable lenses mean that if, down the line you decide to upgrade to a newer/better camera, as long as you stay in the same family of camera, you don't have to replace all your lenses.

Now, the negatives.
They are expensive! figure at least $1000 to get into a basic camera with a few accessories and from there, the sky is the limit.
SLRs are heavy! have a basic DSLR hanging from your neck for a weekend at a convention, and you will be a sore puppy.
SLRs are very complicated, sure, most of them have an automatic mode, but then you are only scratching the very surface of what the camera is capable of.

My kit for my camera (i need to buy some more glass (lenses) soon)
Canon 20d
4 gig microdrive
tripod
assorted filters
18-55 lens
50mm prime lens
external remote (wanted to play with long exposure shots and that as a must)
Photoshop
Lots of HDD space
Quick Release Strap
Gadget bag
Lots of trial and error.

Just like an SLR manual camera, it is a camera in which the lens you see through to frame your image is the same lens the camera uses for taking the picture. What you see is what you get (WYSIWYG).