Overview
Fighters are the smallest, weakest, and fastest class of ship. They are fragile, but their speed makes them nearly impossible to hit with most weapons. They can be the focus of a fleet, a supplemental counter to other fighters, and anything in between. Even if you do not use fighters, you must understand them so your fleet cannot be defeated by the fighter spam strategy. Races with strong fighters include the Tribe, for their increased hitpoints, and the Rebels, with their speed bonus.
Fighter Strategy

Interceptors
Rather than wasting slots on your capital ships for anti-fighter weaponry, a few squadrons of fighters using the escort order can act as interceptors. These fighters should use lasers in most cases, because the minimum range of rockets makes them ineffective in short range dogfights above the cruiser. If you want to use rockets, you should make the range of the escort order higher.
If you are using a race with weak fighters (Imperials, Alliance), you may want to supplement your interceptors with capital ship anti-fighter weapons to even the odds. A strong combo is a few very fast fighters with target painters backed by frigates with anti-fighter missile launchers. Fighters usually attack other fighters first, and the target painter fighters are fast enough to survive for a long time. While the attacking fighters are trying to kill your decoys, the frigates can kill them. As an added bonus, the targeting lasers on your decoys make the the anti-fighter missiles more effective.

Fighter spam
Most opponents have at least a bit of anti-fighter defense, so if you are going to attack using fighters, use enough to overwhelm them. Obviously, this won't work in scenarios with a low pilot limit, although your enemy may be unprepared for a fighter attack, making a few fighters effective.
Fighters have 2 main obstacles, other fighters, and armor. Anti-fighter capital weapons alone are not much of a threat, given sufficient numbers of fighters, and assuming your fighters can overcome the armor quickly.

The first step in a fighter spam strategy is dealing with the opposing fighters.Create an air superiority fighter, either using laser or rocket/painter weaponry. Make sure you have more air superiority fighters than the enemy has fighters. Obviously you may need more/less depending on who you're facing.

You should devote most of the rest of the fighters to the other obstacle, armor. Fighters can operate under shields, however they must still pierce armor, and if there is enough, lasers will be almost ineffectual. The solution is to use rocket and/or torpedo fighters. Torpedoes have extremely high penetration for shields and armor, but may need to engage the shields, depending on the size of the shield bubble. Rockets do not need to engage the shield as often, but they have less armor penetration. .Things to consider when choosing your bombers include race (use torpedos agianst alliance, lasers alone might work against tribe), opposing defenses (rocket fighters are faster, therefore last longer), and fleet composition (rockets and lasers cut through frigates, torpedoes may be necessary against cruisers).

Carriers
Carrier bays are great tools if used effectvely, following a couple of simple rules.FIRST RULE OF CARRIERS- TURN ON CAUTIOUS FOR YOUR FIGHTERS. otherwise they won't use your carrier. The goal is to have a large swarm of fighters doing battle, while the damaged ones slip away and return without being killed. If there are not enough carrier bays, you will have too many fighters waiting, and too few fighters in your swarm to overwhelm anti-fighter defenses. Basically, you should have enough carrier bays. To get a feel for it, click on one of you carriers during battle, and watch the number on the carrier bay. That is how many fighters are waiting for repairs.
There are a few types of carriers, depending on the situation you are using them in.

Interceptor carrier- when using interceptors, you can throw on a bay or two to the ship they are escorting.

WWII style carrier- when you have used almost all your budget on your fighters, and have very little left for your carrier, just make a cheap one, stick it in a corner, and hope your fighters can kill the enemy before the enemy kills the carrier.

Frontline carrier- Strong cruiser that fights on the frontline with your fighters. Nice because fighters have less distance to travel for repairs, but making one that won't just die requires a lot of budget, and other escorts.

Carrier bays are especially effective with the tribe, because their fighters are so resilient.