Many of the buffs and debuffs in World of Warcraft have a dispel type, which appears in the upper right-hand corner of the tooltip box if you hover your pointer over the buff or debuff icon. The dispel type tells you how the effect can be removed. There are four dispel types:
The ability to lay curses on their enemies is almost exclusively limited to Warlocks, although Troll Priests used to have one (Hex of Weakness) before patch 3.0. All shamans gained a curse in patch 3.0.2 through the spell Hex. Only Druids (with Remove Curse) and Mages (with Remove Lesser Curse) can remove them.
These are usually conditions like Rabies, Tetanus, and the like and are usually caused by mobs. However, Priests can cast Devouring Plague, which does have a Disease-type effect, and certain Death Knight abilities cause diseases, most notably Blood Plague and Frost Fever. Paladins (Purify, Cleanse), and Shamans (Cure Disease, Cleansing Totem) can remove Disease-type effects from themselves and their allies. The Dwarven racial ability Stoneform will also instantly cleanse the caster of all Disease effects.
The most common effects are Magic. There are Magic-type buffs (beneficial), as well as Magic debuffs (harmful).
Paladins use Cleanse to remove Magic debuffs from their allies.
Mages use Spellsteal to transfer magic effects from enemies to themselves.
Some Magic-type effects are different, they cannot be removed by any of the aforementioned abilities; these include Banish, Ice Block and Divine Shield. Only a Priest's Mass Dispel can remove these more potent buffs.
Rogues are highly capable of causing Poison-type debuffs and DoTs, but many mobs do as well. Hunter sting abilities are also considered poison debuffs. Druids (with Cure Poison and Abolish Poison), Paladins (Purify, Cleanse) and Shamans (Cure Poison, Poison Cleansing Totem) can remove these. Those with the First Aid secondary skill can create Anti-Venoms to cure poison as well. The Jungle Remedy potions carried by the various spellcasters at Kurzen's Compound in Stranglethorn Vale are also a very useful way to combat poisoning, and finally Dwarves can invoke Stoneform to cleanse themselves of all Poison effects for free once every three minutes.
There are also many lasting effects which do not have a dispel type; there are no general spells that remove these. However, there may still be hope: the ones that impose a specific mechanic upon the target may be susceptible to removal by certain items and abilities that can remove this specific mechanic (for example, Luffa and Stoneform can remove bleed effects, while Tranquilizing Shot can remove frenzy effects). Some lasting effects, however, must simply run their course and expire with time.