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:
Curse
The ability to lay curses on their enemies is almost exclusively limited to Warlocks, although Troll Priests have one (
[Hex of Weakness]). Only Druids (with 
[Remove Curse]) and Mages (with [Remove Lesser Curse]) can remove them.
Disease
These are usually conditions like 
[Rabies], [Tetanus], and the like and are usually caused by mobs. However, Forsaken Priests can cast 
[Devouring Plague], which does have a Disease-type effect. Paladins (
[Purify], 
[Cleanse]), Priests (Cure Disease, 
[Abolish Disease]), and Shamans (Cure Disease, 
[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.
Magic
The largest number of effects have this dispel type; there are Magic-type buffs as well as debuffs. Most spells that have lasting effects cause a Magic-type buff/debuff. Paladins (with [Cleanse]) can remove Magic-type debuffs from allies, while Hunters (with 
[Tranquilizing Shot]), Shamans (with 
[Purge]), and Warriors (with 
[Shield Slam]) can remove Magic-type buffs from enemies. Priests (with [Dispel Magic]) and Warlocks (with the Felhunter's [Devour Magic] ability and the Doomguard's Dispel Magic) can both remove debuffs from allies and buffs from enemies. In addition, the Warlock can remove debuffs with 
[Spellstone].
Some Magic-type effects are of "higher quality" and provide immunity (Banish, Ice Block, and Divine Shield being the most important). These can only be removed by a Priest's 
[Mass Dispel].
Poison
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.
Typeless Effects
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.
Most typeless effects can also be removed using the paladin's 
[Divine Shield] and 
[Divine Intervention], the mage's 
[Ice Block], or the alchemist's Flask of Petrification.