Sure, here goes. If there are any others, just ask!
ILS ==> Instrument Landing System, consists of a localizer (providing horizontal guidance) and a glide path (providing vertical guidance). Allows aircraft to follow a very accurate path to the runway, and is the prefered (non-visual) approach used by most major airports.
VOR ==> VHF Omnidirectional Range, a rather large set of radio transmitters on the ground, which (without getting really technical), allow an aircraft to tune in and follow specific radials either to or from the facility. Typically, there are 360 "radials" with 360 being strait north, 090 being strait east, 270 being strait west, etc. VORs are most often used for en-route navigation (between airports), but they can also be used to conduct approaches when other approaches are out-of-service, or at smaller airports that don't have the traffic or the weather to justify an expensive dedicated ILS system.
DME ==> Distance Measuring Equipment, a system that uses the time a radio signal takes to get from an aircraft, to the DME site, and back again to calculate the distance the aircraft is from the site, and displaying that information to the pilot. These are typically installed with VOR sites, and are paired with them, so when a pilot is tuned into the VOR, not only does he get direction information, he also gets distance information as well. Although they are usually found with VOR sites, they can also be installed separately with NDBs, etc.
NDB ==> Non-directional beacon, basically just a radio antenna that transmits a signal, the same signal in every direction, and equipment on the aircraft just tells the pilot which direction the signal is coming from. Not as accurate as a VOR, but much less expensive to run. A lot of major airports have an NDB about 3-5 miles from the end of each runway, and there are tons of small airports that have an NDB located on the field. Due to the fact that they cost relatively little, require a lot less calibration and testing, and have much longer ranges at low altitude than VOR sites, there are WAY more NDBs then any other type of approach or navigation aid. On the down side, they are not as accurate as a VOR site.