Player Versus Player (PVP) is generally considered any type of combat where the ships fighting each other are piloted by other players and not NPCs. There are many different kinds of PvP combat and many different roles for ships to fill in all kinds of combat in New Eden.
Solo PvP is all about doing everything you can to kill the enemy by yourself.
The objective of small Gang PvP is almost always to destroy enemy ships while having fun. This is probably the most popular type of PvP in New Eden. While there is no direct gain from these ship kills there is an indirect gain in that your enemies are losing money faster than you the more successful you are. There are many different kinds of small gang PvP.
Large fleets are generally saved for fighting for things the alliance needs. This largely all falls under the umbrella of SOV Warfare. In this style of PvP there are usually several fleets involved to avoid confusion. Usually there are two fleets involved, one for capital ships and one for support ships. The capital fleet works to deal damage or protect the objective while the support fleet works to both kill the enemy and protect their cap fleet. Reimbursement of lost ships will usually fall into this category
Wormhole PvP is a special breed of PvP. It is hard to get into wormholes and people live in them for weeks at a time so it is often times good to kick them out of it. Fights are generally a bit harder to find because people do not show up in local chat and directional scan is the only way to find enemies. Ships will generally need to be fit in order to both successfully PvE against the hard content in wormholes as well as defend themselves against enemy attacks.
TEST does NOT live in Wormhole space and generally will not participate in these types of fights unless players from Wormhole space specifically come into TEST space
Faction Warfare is where you sign up with an Empire (Amarr, Caldari, Gallente, Minmatar) and fight with your faction to take over regions oof space located in Lowsec.
You can NOT participate Faction Warfare while you are in TEST as it requires you to join the Factions Corporation
The most famous type of ganking is called Suicide Ganking where people will sit around in relatively cheap ships near popular trade hubs with a scout ahead of them cargo scanning ships as they go by. When a ship with a relatively expensive cargohold is spotted the players will kill that ship letting themselves die to Concord while someone who was not involved in the fight picks up the loot left by the dead expensive ship.