Before jumping into lenses, it's important to note the different naming conventions that each company uses.
Crop sensor cameras tend to have their own lineup of lenses. These are lenses meant to be used with a crop sensor, and are generally cheaper than their full-frame counterparts. However, crop sensor cameras can also use lenses meant for full-frame cameras.
Full-frame cameras should really only be used with full-frame lenses. If you use a crop sensor lens on a full-frame camera, you'll get what's known as vignetting — a dark circle around the other frame of the image.
However, you can technically use a crop-sensor lens on a full-frame body. Many full-frame cameras have a setting that mimics a crop sensor, which will remove the vignetting. There's going to be a loss of image quality if you use this method, however.
Both Nikon crop-sensor and full-frame camera bodies can be used with lenses meant for either system — no matter which type of body you have, you can use a Nikon lens. Canon users aren't so lucky in this regard. Canon's crop sensor cameras will still accept crop-sensor or full-frame lenses, but their full-frame bodies will only accept full-frame lenses. This shouldn't be too much of a problem, but it could be an annoyance if you choose to upgrade to a full frame and find that you can no longer use the crop-sensor lenses you already own.
Nikon's lineup of crop-sensor lenses have the 'DX' label, while its full frame lenses are labeled 'FX.'
Canon's crop-sensor lenses are labeled 'EF-S,' and its full frame lenses are labeled 'EF.'