There are two types of boosters: narcotics/drugs and combat boosters. Since for many pilots narcotics are mostly useless and worthless in current EVE game, when players speak of boosters they usually mean the combat boosters introduced in Revelations.
Narcotics are found in some missions and plexes against pirate NPC factions. They were used in some loyalty points offers (in any case eliminated with loyalty point store), though with their low availabity and difficulty in transportation, these offers were rarely filed.
Combat boosters are made through processing chemicals extracted from special gas clouds which can be found in certain deadspace areas in many regions via exploration. They can be used to increase character's abilities for short duration, most useful during combat, but may have some side effects. More information can be found on the Booster Manufacturing Guide.
Combat Boosters allow you to temporarily boost your combat abilities at the possible cost of random penalties to other attributes. The emergence of Combat Boosters begins with 0.0 COSMOS constellations that contain unique resources, specifically gas clouds that must be harvested to create the boosters.
How to use boosters
To use a booster, right-click the booster icon and select consume. When this is done, you will see the Character Sheet icon in the NeoCom blink. In the Character Sheet, you can see the active effects of the booster and a countdown of the duration left under the Augmentations tab.
Note that you can not consume multiple boosters of the same type.
Side effects of boosters
The boosters usually have possible negative side effects. When you look at the info of the boosters, you can see what these effects are and the likelihood of being affected upon consumption.
This can vary between boosters, but most boosters are illegal and carrying them around can result in confiscation and standing losses.
Note however, that being "under the influence" is not illegal.
Where to get boosters
The manufacturing process utilizes the skill set of mini-professions, and uses specialized starbase structures required to process the gas clouds. The eight new COSMOS Constellations each contain their own unique Booster ingredients, with the more advanced Boosters requiring ingredients from all over the universe.
However, if you don't feel like going through all the trouble of making boosters, you should simply be able to find them on the market when someone else has manufactured them.
To be able to use boosters, you need the following skills: