Minecraft PvP servers (Player versus Player) are usually competitive servers, Minigames servers or Anarchy-style freeplay servers.
Different PvP servers...
Depending on Minecraft servers type, PvP can be main goal or just a way to let players interact with each other. Some PvP servers limit PvP to specific arenas only.
Playing Minecraft PvP servers
You can expect most of PvP servers to competitive, especially if server gamemode is heavily focusing PvP, examples of such gamemodes on Minecraft servers are KitPvP, Factions, Anarchy and even Minigames servers if they provide leaderboeards and track scores.