The term grinding is the act of doing something repetitively, such as killing monsters or working up a trade skill. If one is grinding their levels, it usually refers to killing massive amounts of creeps, or if they grind reputation, they may be collecting items for repeatable quests for a particular faction.
In older MMORPGs, considerably EverQuest, grinding was the primary way to advance your character's level. The quest system was very underdeveloped, as opposed to World of Warcraft's. This is one of the reasons that WoW has received much praise, as the game has less of a treadmill feeling than its predecessors.
Grinding does have some advantages over questing, however. If one chooses to ignore quests for the most part, or even entirely, they have an enhanced chance of finding world drops, which they may either use or sell for large amounts of gold. Also, if one grinds their professions, they can easily surpass the average skill level of their peers and be able to craft much more powerful items. Grinding rep yields substantial rewards (especially with the advent of Burning Crusade), such as Timbermaw Hold or the Scryers.
TIPS: -Before going out to grind levels make sure that you know what your going to kill.
-Make sure you stock up on food, water (if used), amunition (eg: arrows, bulets <if used>)and potions (not necessary but helpfull), and bandages (if used).
-It helps to be rested, but is not needed, (rested is when you earn double the experience for killing a monster).
-Grinding with others will diminish your experience gained per kill.
See also: Farming