PvP, or Player-versus-Player, is a designation for aspects of the game where players are fighting one another directly, rather than trying to defeat computer-controlled opponents. (PvE)
The Horde and Alliance put the war in Warcraft. These two forces have been at odds for a long time, and are unlikely to stop. As a member of one faction, there may come times where you will battle members of the other. It is very rare that you will have an active conflict with a member of your own faction.
When a Horde and an Alliance meet each other, they will sometimes fight or flee. However, this may not always be the case. On a PvE Server, players must flag themselves for PvP by typing /pvp, or taking an aggressive action against a player who is already flagged. On PvP Servers, this will automatically be the case for a player in Contested Territory, which is essentially the entire world except for the newbie zones and Shattrath City. And even if two players are capable of fighting, that does not mean they necessarily will.
Ganking occurs when a player finds a member of the other faction distracted by other business, such as working on a quest, and ambushes them. Typically, a gank leaves the targeted player with absolutely no chance to defend themselves.
Many players will avoid playing on PvP servers because they do not appreciate the constant ganking that goes on, especially by higher-level players that will corpse camp the victim for prolonged periods of time. On a normal (PvE) server, the player can completely opt out of PvP and not deal with this hassle. This has led to all kinds of demeaning terms tossed back and forth between PvP'ers and PvE'ers, the most common of which is Carebear, since both groups often feel that the other does not experience the game the way it is meant to be played.
Players will sometimes band together to perform an assault on an opposing town. This has little direct purpose, other than the sheer thrill of it. Less frequently, a raid may organize to attack an opposing faction's leader in one of the capitals.
The side effects of a town raid are that vital NPCs are often killed, barring the progress of low-level players. This can also provoke members of the defending faction to retaliate and spark open conflict. This is often how PvE servers will start PvP conflicts, and was much more frequent before the inclusion of battlegrounds.
A few zones have outdoor objectives for PvP. While there is no need to participate in these objectives, even while in the zone, they provide a zone-wide bonus to players that are there. For example, by controlling all three towers in Hellfire Peninsula, players will do 5% more damage in combat.
Outdoor objectives exist in Eastern Plaguelands, Silithus, Hellfire Peninsula, Zangarmarsh, Terokkar Forest, and Nagrand. The new zone of Wintergrasp in Northrend will be almost exclusively for outdoor PvP conflict.
The Honor System provides a means for rewarding players who participate extensively in PvP. By killing an opposing faction member, they will earn an Honorable Kill or (HK) that gives them a small amount of Honor. There are select PvP vendors that sell high-quality equipment for Honor points.
Honor can also be earned for participating in a battleground. Honor will not be earned for killing an opposing faction member that is "gray" to you. A level 70 character will not earn honor for slaughtering level 1's.
With Wrath of the Lich King, the honor gained in Battlegrounds and Lake Wintergrasps was greatly increased, and so were the cost of the new honor item, some going to three time as much honor as their previous slot items. The maximum honor point one can hold however, has remained at 75,000.
The Honor System started out with some of the same pretexts it operates under now, but with players earning PvP ranks instead of honor points, and a certain rank was required to purchase equipment from the vendors. This system was ultimately done away with because it was restrictive to the number of players who could obtain a high rank and because it forced the top players to play excessive amounts of time to acquire and maintain their rank.
The old honor system also penalized players for killing civilians by giving out a Dishonorable Kill or DK. If enough DKs were given, the player could even lose their rank.
Battlegrounds (BG) are mini-games that use PvP. Players queue with a Battlemaster to sign up for one of these games, and when enough players from both sides want to participate, a match is created. Note that potential players for a battleground are drawn from several servers, called a Battlegroup.
Each BG has its own goal, resembling typical strategy games like Capture the Flag and King of the Hill. The number of players required for each BG is different, ranging from small-scale skirmishes of 10 to full-out assault of 40. Battlegrounds reward with honor, and there are also Quartermasters that sell items to players of a certain reputation level.
Twinks are extremely geared players that exist primarily low levels. Originally battle grounds brackets were 10-19, 20-29 and so on. So twinks would level to 19 or 29 and stop. Blizzard eventually nerfed the twink by adding experience to the battle grounds making a character eventually level to the next bracket.
They followed up this by adding a way to turn experience off. They also added two queue types for battlegrounds, one for experience on and one for experience off players(twinks). The main issue is there was far more players with experience on and very few twinks on a lot of battle-groups. Some still exist where a twink can get into a battleground but it is rare.
Today there are new adventures for the Twink since the invention of heirloom weapons and gear. Also Blizzard shrunk the brackets to every 5 levels so (10-14, 15-19 and so on). This allows players to gather tons of "best in slot" gear with a high level character and give it to a low level character. Keeping experience on is easily done in the level 10 to 14 bracket because once the character levels to 15 they can pass their gear back to the high level characterful and delete the twink. It does not take much time or gold to level back up to level 10 and start owning the 10 - 14 bracket once again. the correct gear needs to be acquired and can be seen here for each class and spec at the 10 through 14 bracket 
Duels are single player challenges. While dueling is technically PvP, many players do not treat it as such because the battle is pitched, in a very limited space, with little room for the element of surprise. And, as Blizzard has reminded us repeatedly, the game is allegedly not balanced around 1 vs 1.
Teams of 2, 3 or 5 can participate in Arena conflict against other teams. Although an arena battle is similar to a duel in that the battle is pre-arranged and be against players of the same faction, there is far greater flexibility with game dynamics with more players and much greater space available.
Arena teams have a score based on their performance. Winning teams earn higher ratings, and are paired against other teams that have similar ranking. Opposing arena teams are also taken from within your battlegroup. Successful participation on an arena team will earn points toward a loot system similar to honor, but independent.
Put quite simply, this is a new PvP-centric zone in Wrath of the Lich King. Lake Wintergrasp is not instance, is always available and is filled with npcs, mineral veins and herbs. There's several quests in there that involves the siege of Wintergrasp Fortress. Every 2 hours 30 minutes, the siege occurs and members of the Alliance and Horde must go and fight.
Most information for this page has come from WowWiki