Warsong Gulch is one of the easiest and shortest pvp battlegrounds in World of Warcraft. It consists of 2 sides the Alliance and the Horde. There are 2 “bases” in this battleground that contain a flag inside. Ten players are on each team and the goal is to bring the other team's flag into your base without having yours being in the possession of an enemy player. To win the game one side needs to get the enemy flag and “score” 3 times. The game is essentially “capture the flag” and it doesn't matter how many kills you get although it helps go toward honor points. Here is a map of the battleground:

Arathi Basin
This battleground is a more complex and usually more time consuming pvp battleground than warsong gulch. First of all there are still 2 sides the Alliance and the Horde who both start off in their bases. There is a maximum of 15 players on each team. The object of the game is to obtain and hold different “stations” in the battleground. There are 5 of them and they are the farm, blacksmith, lumbermill, mine, and stable. Usually a team aims for obtaining and holding 3 of them to out do the other team who can only get 2.
As time goes on there is a point system that has a goal of 2000 points and the team that gets to 2000 wins the match. For each “station” that a team holds points are awarded to that team gradually. So for example if one team had 3 stations and the other had 2 more points would be awarded to the team with 3 gradually. However through the game stations will be lost and won. The stations can constantly change hands. Here is the map for Arathi Basin:

Alterac Valley and Eye of the Storm
I have to be honest here and say that although I know a lot about the first to battlegrounds I really don't know about the last 2 which are Alterac Valley and Eye of the Storm. I have not had the chance to have my guy get up to the required level for these battlegrounds. However, I do have maps of them:

Here is the map for Eye of the Storm:

The Rewards:
You get honor point when you play a pvp battleground and when you kill an enemy player. You get more honor when you win a match and less when you lose one, however everyone still gets honor points. The reward for playing is essentially for honor points to buy pvp items like armor sets and weapons. In addition however, you get a medal for playing the battleground. You get 3 medals when your team wins and 1 if your team loses.
Arena
The arena is supposed to be a step up from the “traditional battlegrounds” since you have to be level 70 to play.. There are three Arenas in the game. The Ring of Trials can be found in Nagrand, the Circle of Blood is in the Blade's Edge Mountains, and there is the Ruins of Lordaeron. There is a ranking system and honor reward for this. There are different arena seasons as well as arena titles. There are arena prizes and special prizes that you can purchase from an arena vendor. There is also singular and team based arena.