Full Frame vs. Crop Cameras
Most brands of DSLR came in two sensor sizes: full frame and crop. A camera that is "full frame" (Nikon full frame cameras are called FX) has a sensor that is the same size as a frame of 35mm film. Crop sensors are roughly half that size: Canon crop cameras have a 1.6x crop factor, and Nikon crop cameras (DX) have a 1.5x crop factor
Canon Full-Frame Cameras:
1D series (excluding the 1D, 1D Mark II, 1D Mark III and 1D Mark IV, which were 1.3x crop)
1Ds series (discontinued as of the 1D X, 2012)
Nikon FX (full frame) Cameras:
Dxxx series (e.g., D800, et. al.)
Canon Crop Cameras:
xxD series (e.g., 60D, et. al.)
Nikon DX (crop) Cameras:
Dxxxx series (e.g., D5000, et. al.)
Dxx series (e.g., D40, et. al.)
Full Frame at 17mm (Nikon - FX) 1.6x Crop Factor (APS-C) at 17mm (Nikon's DX is a 1.5x crop factor)
IMPORTANT TO NOTE: the photos above illustrate how the same lens will show two different views depending on if it's on a full-frame or a crop camera. The photo on the left was taken using a 17mm lens on a full frame camera. The white box - which matches the photo on the right - shows what the EXACT SAME LENS on a crop-camera would capture due to the sensor being smaller. The lens doesn't change sizes, the area of the image that's captured from the lens changes based on sensor size.
ALSO IMPORTANT: Canon and Nikon both make lenses designed or use with crop cameras. Canon crop lenses are called EF-S (think 'S' for 'small sensor') and Nikon crop lenses are called DX lenses. Nikon DX lenses will work on Nikon full-frame (FX) cameras (but it doesn't make sense to buy them if you have an FX camera because you'll only use the middle of the sensor) but Canon EF-S lenses WILL NOT EVEN MOUNT onto full-frame Canon cameras. Conversely, all full frame/FX lenses will fit on crop camera, you'll just be losing a lot of the lens' coverage as illustrated above.
When it comes to buying more equipment for your camera don't hesitate to buy off-brand to save money, except for when it comes to lenses. The lens is the single most important part of your camera: if you use a bad lens, it won't matter how great your sensor or autofocus system is. If you're shooting a Nikon camera - buy Nikon lenses. If you're shooting Canon - buy Canon lenses. Both companies are primarily optics companies, and that shows in their lenses. If you have good reason to buy off-brand (or if you want to drop a fortune on a beautiful Zeiss lens) then go ahead, but as a general rule it's much more worth it to get lenses that match your camera manufacturer (especially with Canon and Nikon).
Prime Vs. Zoom
Simple: prime lenses are a fixed focal length (e.g., a 50mm or 100mm et. al.) and zoom lenses have a range of focal lengths (e.g., 28-105mm, 70-200mm et. al.).
Prime Lenses: because there are less moving parts and specifically made for a single focal length, they tend to be much lighter and optically sharper.
Zoom Lenses: They cover a wider range of focal lengths - less lenses to carry around
Ultrawide lenses are any lens that have a focal length 20mm or smaller but are not fisheye lenses (see below). Note: ultrawide lenses should not be used to try and get more in the photo: they're used to get up close and personal with your subject matter like when I shot these flowers with a 14mm lens.
Fisheye lenses are lenses with an intentionally rounded distortion (between 5.6mm-15mm). NOTE: some fisheyes are so wide that they make a circular image on your sensor. Canon makes an 8-15mm zoom fisheye lens that on full-frame makes a perfect circle at the 8mm setting and fills the frame at the 15mm end.
Lenses roughly 24-35mm
Lenses roughly 40-60mm (they're called "normal" because it's about what the distance that humans see with our eyes. If you're staring straight ahead and then put up a lens with one of these focal lengths in front of you, you'll see everything at roughly the same size. 50mm lenses are considered the most 'normal' focal length.
70mm + lenses.
Macro (Nikon 'Micro') lenses alllow you to get super close-up images. A 1:1 macro lens set to 1:1 is like setting the image sensor on the object you're photographing directly: see an example here of macro photos of eyes. Macro lenses are amazing for detail photos or photos of small objects like jewelry, insects, etc. Macro lenses can be 50mm, 60mm, 100mm, 105mm, 180mm, etc. The longer the focal length (higher the number) the further you can be away from the object that you're photographing at 1:1. Note too that to photograph at1:1, you'll need to set the lens to "manual focus" and move the entire camera back and forth to focus it. If you're using a tripod you can make use of macro rails to help out with this.
Some people refer to 85mm prime lenses as 'portrait' lenses but that's certainly not etched in stone. There are two basic criteria for choosing a good portrait lens. First, use a prime lens - primes are much sharper than zooms as a rule, and sharpness is very important for portraits. Second, use a somewhat longer focal length: 85mm at a minimum. If you're right in your subject's face they're not going to be as comfortable as if they feel like they have their space.
How many lenses do I need?
Here's the good news: not many. Today's lenses are made to excellent standards, and you don't need to tote around 300 lenses to get good images. The fewer lenses you have, the better in fact. If you want to cover all the bases, have one ultrawide, one normal, and one telephoto lense - all three can be zooms, all three can be primes, or a mixture. When I'm out shooting 90% of the time I use only my 17-40mm lens on my 5D Mark II, and 90% of the time that I'm using that lens I shoot it at 17mm. That's me - you might have a much different preferred focal length. If I do bring along another lens, it's my 100mm macro: those two lenses alone generally cover 99.9% of what I need. If you're just getting into DSLR photography the kit lens that came with your camera (most likely an 18-55mm zoom) is all you need for now - don't feel like you need "better and more" lenses to take excellent photos.