The following article was written with the 3rd ed codex in mind. A lot of the general stuff should still apply, but the more specific bits are out of date. I will revise this page, but I am actually writing a more in depth document also, and that may simply replace this in time.
Eldar cannot survive a battle of attrition
No unit is survivable on its own
Back your units up
Combine your tactics
Use your speed
Separate your opponent's army
Target isolated units
Pay great attention to deployment
Be aware that the Eldar codex provide four categories of units
Fodder
Close combat specialists
Anti-tank
Supporting fire
Examples:
Scorpions are not effective if they are forced to march across the board on their own. Far better to send a squad of storm guardians with them, or alternatively, mount them in a wave serpent. In this situation the wave serpent can be thought of as one of their supporting units. It's primary purpose is to improve the ability of the scorpions. Both the wave serpent and storm guardians can be thought of as support units, while the scorpions are CC specialists.
Depending upon how they are armed, Falcons can be Anti-tank, or support.
If anti tank, they are probably armed with a bright lance or missile launcher -
thus can only move 6" to fire all weapons. In this situation they should be
treated as a fairly static unit. Forget about moving more than 6", find some
cover and stay hull down. Then, importantly, stay absolutely still so long as
you still have targets. This way when you receive the inevitable immobilized,
you can continue to fire.
If chosen for a supporting role, they are far better off being armed with a
weaker gun that has more shots. This is when you might consider upgrading the
shuriken catapults also. In this role they should be providing supporting fire
to your fodder and CC units. Getting up close with the enemy, and staying 'in
his face'. Any shots at the Falcon are shots not taken at the important CC
units.
Support is a fairly wide ranging category. In some situations, anti tank can
be thought of as support. A dedicated support unit is (obviously) the support
battery. I'm slowly coming to the conclusion that you should always have at
least one support battery, even of a single gun. A couple of shadow weavers can
force your opponent to move. If your opponent is a defensive player, this will
upset them greatly. On the other hand if you are dealing with an offensive
player, you would probably be better off with a D-Cannon, as its shorter range
should not be a problem.
An undedicated support unit is the guardian defender squad. In many situations
they will take the double role of fodder and support. If primarily chosen for a
support role, they should be armed with a long range weapon (missile launcher)
and set behind the rest of your army.
The Wraithlord is a brilliant example of a fodder unit. Often it will completely fail to get into combat. Therefore it should almost always be given a gun. The un-initiated will waste tons of firepower on wraithlords that would be far better used against more dangerous targets - like the falcon or wave serpent. I recommend the lower rate of fire weapons. If you're going to have a BS4 unit, it may as well be put to good use.
Here is a breakdown of the army list by category:
Avatar - CC, fodder (yes fodder. Sad but true)
Farseer - support
Warlocks - support
Striking Scorpions - CC
Howling Banshees - CC
Fire Dragons - Anti Tank
Warp Spiders - support
Dire Avengers - support/fodder
Storm Guardians - fodder
Defender Guardians - fodder, support
Rangers - support
Wave Serpent - support
Guardian Jetbikes - fodder (again, sad but true. They take a lot of punishment,
but save the rest of the army)
Shining Spears - Anti tank (BS5 Brightlance! I can't think of any other role you
could want them for)
Vypers - support, fodder, anti tank (if armed appropriately)
Swooping Hawks - support (and CC with the right exarch, but that's not fully
appropriate)
Falcon - support, anti tank.
Support batteries - support
Dark Reapers - support
Warwalkers - support (mainly. See vyper)
Wraithlord - fodder
Fire Prism - fodder (seriously, don't bother they never hit!)
Vypers and war walkers are tricky . If they're armed with anti tank weaponry, they are far more anti tank than support, but they still manage to be fodder. Vypers are cheap, and thus loosing them means other units are going to survive. I do not recommend upgrading them in any way shape or form.
Once you have a good understanding of the above, you can begin to master the more complex strategies. The simplest example being the old Patrol mission. If you only have one small unit of guardians on the board, and your opponent has heavy bolters. Don't leave cover! Stay as hidden as possible. It is simply not worth moving - you will lose a unit for absolutely no gain.