Eye of the Storm has two 15-person teams. The battleground consists of four towers, one located at each of the four corners, and a flag in the very center. The battleground itself is located up in the sky; don't fall off the edges or you will die.
Each team starts at 0 points, and the first team to reach 2000 wins. Points are gained in one of two ways: controlling the towers, and capturing the flag. Capturing the flag earns roughly 100 points. Controlling towers earns points based on the number of towers controlled and length of time you've controlled them.
Each team starts the game on an elevated platform at opposite ends of the battleground. It requires some care to jump down from this platform with no damage; it requires no care if you don't mind a damage hit up to 50%.
Hint: Learn the tower names and locations relative to one another so it will be easier to know where to go when someone calls for help. When in doubt, use the map to determine which tower is which. When communicating with in-game chat, players will usually abbreviate the tower names. The two towers closest to the Alliance players are called the Mage Tower (MT) and the Draenei Ruins (DR). The two towers closest to the Horde players are called the Blood Elf Tower (BE) and the Fel Reaver Ruins (FR).
To control a tower, you simply have to have more of your players present at that tower. When you arrive at a tower, there is a slider bar that appears under the mini-map. The slider indicates whether the tower is horde-controlled or alliance-controlled, and to what percentage. The more time players from a faction spend at the tower, the further over the slider will go.
There is a gray area in the middle of the bar that represents "uncontrolled". Uncontrolled towers are not earning points for either team, nor can they be used for flag captures. Tower statuses can also be seen on the map.
Players pick up the flag in the center of the battleground by right-clicking it. When the flag is returned to a controlled tower, it automatically "captures" as soon as the player runs over the flag drop area. (It is located at the top of the steps for each tower, and looks like a gear embedded in the ground.) After the flag is captured, it is automatically reset in the center position, where it can be captured again.
You can only cause a player to drop the flag by killing him. Once he is dead, the flag drops at that location. The first player to right-click on the flag (from either team) will become the new flag carrier.
When you die, you resurrect at the nearest tower controlled by your team. If your team doesn't control any towers, you resurrect back on the starting platform.
Strategies and Tactics
Controlling the towers is considerably more important than capturing flags. For one thing, if you don't control any of the towers, you can't capture the flag at all, because the flag must be returned to a tower that your team controls.
Additionally, you will gain victory points far faster by dominating control of the towers. In fact, if you manage to control all four towers, you will win by simply maintaining that control for about a minute. Continuous control of three towers is also usually sufficient for victory, regardless of what happens with the flag. (The logic is simple: you are earning points at a rate of 3:1 over your opponents.)
The best strategy is to go for three towers right at the start. Bad teams/players will run directly into the center and capture the flag, which is pointless for the reasons mentioned above. An early flag capture may give a team an early lead at the expense of long-term strategic control of the towers, and probably won't last.
When the game starts, it should be sufficient to send two players to each of the two closest towers and complete their capture. (To clarify: that's four total players, two per tower.) Those players should then remain in defense of those towers, and call for help as needed. The remaining players should descend on one of the towers that will no doubt be occupied by players from the opposing team. In a well-coordinated effort, this shouldn't be difficult to do… unless the other team is also well-coordinated.
Having controlled a third tower, you'll have a choice to make. If the enemy is showing poor offensive capabilities, get greedy and go for the fourth tower. Leave two defenders at tower number three. The remaining nine can head to tower number four. Again, this will most likely be an overwhelming force, except in the case of an opposing team that is equally well-coordinated.
If the opponents are showing good coordination or are simply more powerful, it may be impossible to retain control over all four towers. In that case, do everything you can to retain at least three, then send a group in to capture the flag. Repeated flag captures at this point will do two things: get you closer to victory, and keep some percentage of the enemy forces busy trying to stop you.
Above all else, keep in mind that retaining control of the towers is the single most important goal.
You may want to send a couple of people in to capture the flag sooner, if for no other reason than to give the opposing team something else to deal with. Bad teams (i.e. most of them) will be very upset and send a lot of forces to stop the flag carrier, thus leaving their towers wide open for attack. A couple players can essentially sacrifice themselves with the knowledge that it is furthering the cause of the team. (Luckily battlegrounds don't cause durability damage on your gear, so you can die as often as is necessary.)