In order to be successful in arenas you need to have an extensive knowledge base.
It is important to have talents and abilities of other classes memorized, as well as each class spec's strengths and weaknesses.
Vital information about a class/spec include cooldowns (CDs) available, cooldown duration, and knowing certain class "tells" when they are about to do damage.
Great players are able to track both opponent cooldowns as well as use of their PvP trinket.
The biggest key to success in arena is communication. Voice chat is not absolutely necessary, but is definitely preferred. Call out heals that are going out, crowd control (CC) that is going out, CC that is coming in, damage taken, kill opportunities, etc. If your enemy is focusing on one person on your team and they quickly switch to another, this information is vital to the healers. If you are splitting damage and one of the targets gets low in health, this must be shared with your teammates so they can focus fire on that target and CC the other team's healers.
In the arena, you must always know your surroundings. Know where to line-of-sight your enemies, and know where they can line-of-sight you. Know where your enemies are going and know where your teammates are going. Raid symbols can often be a huge help to track movement.
You must find a perfect balance between doing damage, keeping aware of your surroundings and not getting tunnel vision.
Save your burst damage for the right time. Have your cooldowns ready, which includes class abilities, crowd control, and trinkets, and wait until diminishing returns are stacked up on your opponents' CC. When you feel like the pace is slow, CC whatever you can on their team (this ranges from long-term, such as poly and fears, to short-term, such as interrupts and incapacitates). Pop your cooldowns and trinkets, and burn as hard as you can. If you cannot get a kill shot, try to recover, stabilize your play, and control the match until you have your cooldowns back up.
Be fluid when it comes to picking a kill target. Many teams will go on classes such as a warlock or mage because they are known for taking more damage, but these classes have adapted to this. Mages are used to being targeted so they are good at running. Warlocks are used to getting beat on so they gear and spec to take a beating.
Don't forget that you can bind all of your skills using the "Keybind" option in the game menu.
Strategy:
The more you play in arena, the better you will be able to quickly recognize the opposing team's composition and know your own team's strengths and weaknesses against their specific makeup.
Coordination between teammates is important, especially when quickly switching targets to exploit an enemy player that has gotten out-of-position or used their PvP trinket/large cooldown at a bad time.
Practice with your team and develop complex strategies.
Again, the more you play together the better your reactions to enemy action will become. Just as in any other battle situation, having the ability and knowledge to counter an opponent's attack is crucial to your success.