Minecraft wiki entry for Server, associated text: Survival: These servers are servers that use only survival mode. They are similar to Anarchy Servers, except there are usually rules, and usually prohibit Griefing and Exploiting.
Creative: These are servers that only use Creative mode. Usually, servers give the players Creative, but limit the items they can access and/or limit the plot size. This is usually done to prevent griefing to other players' plots and to prevent excessive lagspikes.
Minigame: These are servers that host multiple minigames, ranging from Spleef to Capture the Flag.
Anarchy: These servers have no rules at all. Many players engage in cheating, exploiting, and use of vulgar language.
Faction: Similar in concept to Survival servers, but claims are used to mark land as "yours". Untrusted players will not be able to build on "your" land.
Personal/Private: These are servers that players can make on their own. Usually, the link is private, shared only with the user's close friends.
Hardcore: The same as survival mode, except players only have one life. This typically ends with a "last man standing" scenario, This type of server is mainly used in SkyWars and Survival Games.
Roleplay: These servers have their players acting as their own persona, sometimes being dictated by the Roleplay that is happening.
Pay-to-Win: These servers have become slightly uncommon, as it violates Minecraft's EULA, but Pay-to-Win (P2W) servers (occasionally known as Pump-and-Dump servers, which are similar in concept to P2W servers) are designed around allowing users to purchase advantages (likely in the form of Ranks) for real-world money.