Training the skill thermodynamics unlocks the ability to heat your modules. This is one of the most important pvp skills. Proper heat management makes your ship hit harder, fly faster and tank more. I would recommend training thermo to 3 or 4 as soon as you can. For those of you who want an indepth mechanical guide, Kadesh has a post at https://forums.eveonline.com/default.aspx?g=posts&m=341573 . Kadesh is the person behind pyfa. He probably knows more about eve mechanics than anyone else, including the devs.
Heating a module either increases its range, power or decreases its cycle time. When a module reaches 100% heat damage it goes offline until you dock up and repair it. Make sure you turn the module back online after repairing. If a module is partially damaged, you can repair it with nanite paste. There are 2 skills for nanite paste, which decrease the time needed and the amount of paste needed. Frigates can do with less than 50 nanite. Ships with large guns might need several thousand nanite over the course of a roam.
Note that as of 5/2014, repairing with nanite paste is very buggy. Sometimes if you repair a stack of guns, only one will repair. Or sometimes they will all start to repair, but only the first will ever complete. Or sometimes they will all repair but until you unlink the guns and reload them all only the first will fire. Other times, the repair will just fail and the module will remain with heat damage.
Heat damage spreads to all modules in a given rack. Heat spreads more often to adjacent modules. Putting your mwd and your shield booster next to each other means they will put damage on each other. If you put a less essential module in between them, you are less likely to burn out your mwd by heating your shield booster. (And less like to burn out your shield booster by heating your mwd). Make sure you can see your passive modules. People die from burning out their tank mods without realizing it.
This post is a guide for heating modules slot by slot.
1) Weapons - Heating weapons increases damage dealt. On some weapons it increases the volley damage, on others it makes them cycle more often. Don't waste your highslot heat when your guns are not hitting. A final note on guns - slowly cycling weapons (particularly artillery and light missile launchers) release incredible amounts of heat. A destroyer fit with these weapons can burn out its entire rack in as few as 2-3 cycles.
2) Neuts - Heating neuts decreases their cycle time. Heating a neut is a good way to put more pressure on an active tank, or break a tackle on you. Make sure you are not wasting your heat on a cap empty target. Also, when using neuts, you almost always want to counter cycle them. If you have 2 neuts with 6s cycle time, sequence them so that one is firing every 3s. Neuting all at once in a panic is a great way to waste all your capacitor - a single cap booster or nos will then counter all your neuts.
3) Nosferatu - Heating a nos decreases cycle time. Sometimes heating a nos makes it less useful to you! If a nos has a 5s cycle time, and your scram has a 5s cycle time, timing your nos just before your scram means you cannot be neuted off. If you then heat your nos, it will fall out of phase.
4) Smartbombs - Heating smartbombs decreases cycle time. They are here for completeness, you do not heat them often.
5) Remote shield boosters and remote armor repairers - Heating them decreases cycle time. Especially for armor reps, heating the first cycle can save your target. Armor repairs apply at the end of cycle, so heating them makes the first wave of hp boost come faster.
1) MWD - Heating MWD increases the speed boost. A well timed MWD heat can be the difference between tackling a target and missing it, or the difference between making it back to a gate after jumping into a gatecamp and exploding. MWDs release a lot of heat, particularly on frigates - frigate mwds can burn out after 2-3 cycles.
2) Afterburner - Same as mwd. Heating an afterburner as you orbit closely lets you mitigate more incoming dps, but may also affect your own damage application. Afterburners can be heated longer than microwarpdrives.
3) Scram/Web - Heating these modules increases their range. This can let you tackle something that you would have otherwise missed, or let you keep someone tackled a little longer as he pulls range on you. In particular, take the situation where you have a scram and mwd, and the target has a scram and web and is slowly pulling away from you. If you heat your scram and he does not heat his, there is a small window where you can pulse your (heated) mwd and close the distance again. Don't waste your heat on your scram/web if the target is well inside tackle range. When using a scram or web as a defensive module, it should be in a heated state by default. These modules can be heated for a long time.
4) Warp Disruptor - Heating this module increases its range. It works the same as scram/web, but a disruptor gives you no range control. It can be used to point someone who would have otherwise warped out, or give you a little more range on a dangerous ship. It can also give you enough time for that 1 extra killing shot. This module can be heated for a long time.
5) Shield booster - Heating this module increases its boost amount AND decreases its cycle time. It greatly increases your active tank. Fitting a shield boost amplifier increases the heat released by the shield booster. Heating your shield booster increases its cap efficiency, so when you are pulsing shield booster under neuts, always heat it.
6) Ancillary Shield booster - Heating this module increases its boost amount and decreases the cycle time. By heating an ASB, you increase the total hp it gives before reload. You should almost always heat this module.
7) Active shield resists - Heating this module increases the resistance given. Heat it to increase your ehp, or your effective shield boost amount. Don't heat the wrong hardener though - its a waste to heat an EM ward when you are taking only thermal damage. Heating your hardeners makes it easier to manage your shield boosting, as your shield ehp is increased. This is particularly helpful on cruisers with x-lasbs.
8) Ewar/Counterewar - Heating ewar (tracking disruptors, sensor damps, ecm, as well as eccm, remote sensor booster, sensor booster, etc) increases its effectiveness. They release a decent amount of heat. Because of how eve math works, heating ewar is incredibly effective. Two heated TDs are twice as powerful as two unheated TDs.
9) Cap booster - Heating a cap booster decreases the cycle time. This gives you more capacitor. When you are trying to run a shield booster, and can only manage a single cycle per cap booster, heating the cap booster is like a 20% increase in your tank.
10) (Omnidirectional) Tracking Computers and Target Painters- Heating this module increases its power. Heating this gives you either more range or tracking. Extra range or tracking means extra applied dps, so you would heat this when you are not applying full dps and want more. Target painters are included here because they behave almost identically, except they apply the bonus for everyone shooting your target.
11) Hull repairers - Again here for completeness. Don't heat these, don't fit these.
1) Armor reps, armor hardeners, ancillary armor reps - The heat benefits are the same as for shields. Heating a single armor rep is more cap efficient than running both armor reps without heat. With an ancillary armor rep, you again want to heat it as much as you can (before it runs out of paste).
2) Reactive armor hardener - Heating this module decreases cycle time. As of 5/2014, heating this module increases cap use as well, but that is being changed in the Kronos expansion. With a faster cycle time, this module adapts to incoming dps faster. There is no benefit to heating a fully adapted reactive hardener.