Paladin Introduction
Paladins in World of Warcraft are part warrior, part healer. They are the holy knights of mythology, with the skill to deal death to their enemies the battlefield and the ability to heal their allies. They are one of the 3 main "hybrid" classes, along with Druids and Shaman, and many consider the Paladin and Shaman to be direct nemeses. Depending on how these warrior healers distribute Talent points, a Paladin can be a strong tank, reliable healer, or offensive dps, without losing the flexibility of handling all three roles to some degree regardless of specialisation.
Like the Paladins of other fantasy worlds, the Paladins in WOW are also the sole class of characters with the ability to repel and damage Undead more effectively than other classes.
(Note some of this information might be outdated since I'm only basing this on my own personal knowledge, about 1.5yrs old now ;).
 Playing a Paladin Effectively
Paladins are extremely versatile as long as the player does not force the character into roles they are not suited for. Paladins can make decent Tanks, become great main tank healers, and can also deal a significant amount of damage.
What distinguishes a superb Paladin from a mediocre one is the recognition that a Paladin is most strong as a team player utilising their hybrid skills, not as a soloist only focussing on a single aspect of the character.
Retribution and Protection Paladins can still heal, and Holy Paladins should help tank and dps (or not, as the situation dictates).
In higher level instances a Paladin's role might be more restricted (to that of a healer, or secondary tank for example), but in almost nearly any role, no matter how restricted, a paladin must be aware of his surroundings and be ready to apply his OTHER capabilities as circumstances change.
 PVP Paladins: Team Players FTW
In PVP, a Paladin is a true force to be reckoned with - not because they can do massive amounts of damage by themselves or single handedly take out players with ease, but because their multi-faceted usefulness allow them to amplify the effectiveness of just about any other character class they are grouped with. A warrior - paladin combination is absolutely devastating, as are hunter - paladins.
A well prepared Paladin who looks out for his teammates is far more valuable than one which only cares for personal glory. The key is to balance offensive, defensive, and healing tasks to maximize both your effectiveness and your team mates. In PVP especially, the name of the game is to outsurvive the enemy, and Paladins are amazingly tough to defeat, so often the Horde will focus on the other team mates, whom you can support by healing and protecting.
 Sample Strategy: Hunter - Paladin Combo
Using as an example, a Paladin-Hunter combo, the typical response of enemies is to attack the Hunter (arguably the more DPS capable class depending on specialisation). This is a PERFECT opportunity for you because the Hunter is agile, can lay traps to slow the enemy, and also attack while trying to stay out of damage range. Meanwhile the Paladin can heal and also do occasional DPS as possible. Hunters are also supremely effective against cloth based classes (healers and mages), which make this combination quite devastating.
Should the enemy attack the Paladin first, in the hopes of incapacitating the healer in the team, the situation is STILL ideal. The hunter can DPS the weaker opponent while the paladin focuses on protecting and healing himself. Use stuns to disrupt the enemy spellcasters and healers. When your health becomes critical, use your Divine Shield. This effectively wastes your enemies time because they have spent their entire time trying to kill you while both you and your team mates have been concentrating all efforts on one of their weaker members. While in the "bubble", heal yourself but watch your team mate, if they start getting hurt switch your healing to them instead. Even if the bubble expires your enemies are now faced with 2 annoying decisions: continue trying to kill the Hunter, who is constantly healed by the Paladin, or switch back to the Paladin who might still be hurt. By this time you might have gotten pretty close to kill one of the enemies. The minute you can unbalance the two sides you are on your way to victory.
 Pros and Cons
 For tanking
Protection paladins are THE BEST aoe tanks in the game. There is no argument. They have the ability to hold the aggro of many, many mobs. If there gear can withstand it, they can tank numberless monsters.
 For healing
Holy paladins are one of the best single target healers in the game. Flash of light, holy light, and holy shock work together perfectly to keep a player at full health at all times. With spell crit directly affecting their heals, they can crit for enormous numbers. They are the only healers that wear plate so they can survive a few hits unlike priests and druids.
 For tanking
Because they are ideal for aoe tanking, paladin tanks are purely situational. Warriors and druids are much better tanks for bosses as they are perfect for single targetting mobs. Protection paladins are usually used for tanking multiple trash mobs or adds in a boss encounter.
 For healing
Holy paladins aren't very good at raid healing or healing multiple players. A priest can throw renews and flash heals to get the job done. A druid just throws up a few HoTs. A shaman can chain heal multiple players. But because flash of light heals for very little, they cannot heal multiple players up as fast as they would with one player.
Related ThreadsPaladin control - last post by @ Oct 22, 2014
Paladin replays - last post by @ Nov 4, 2006
Paladin v.s. Death Knight - last post @ Jul 8, 2014