There are numerous Types of Minecraft Servers available:
This type of server usually prevents most or all attempts to build or destroy blocks. Instead of traditional gameplay, these servers offer unique gameplay elements, often inspired by Role-Playing Games. These servers are rarely unmodified, often bearing significant changes via server modifications which make normal gameplay impossible.
Capture the Flag
This type of server offers a new type of gamemode for a server. However this usually requires proprietary software. Two teams are chosen and they battle it out, trying to collect the other teams flag and return it to their home point a number of times to win the game.
Typically consisting of a premade map and the survival game mode, this server offers a variation on the survival mantra such as surviving without the use of a certain material or in unique terrain. Describing a Challenge server is best accomplished by example - Skyblock servers, one type of challenge server, place players on a small island with exactly enough renewable resources to survive and expand the island.
This type of server offers one or more areas, usually protected from griefing using proprietary software, in which players can recieve a space to build in, either for free or for some form of currency system as in an economy server. This type of server is rarely unmodified.
This type of server is played entirely in the creative game mode, with building being the focus of the community. Rarely are these servers unmodified due to the ease with which griefing is accomplished; most of these servers use proprietary protection systems to prevent players' constructions from being modified without their consent.
This type of server is modified to introduce a currency system, usually in an intangible form. Players trade in this currency to obtain resources, or hoard it in an attempt to be the wealthiest person on the server. This can be included on several other servers.
This type of server is modified with proprietary software to allow players to form groups called Factions and claim & protect land from other Factions. Usually this is accomplished with the Craftbukkit server software and the Factions plugin. Land protected by a Faction cannot be built on or destroyed by players belonging to a different Faction. Factions can also ally with each other or declare war on each other. The amount of land one Faction can protect at any given time is decided by the number of players belonging to that Faction. The FactionPvP servers usually allow griefing and are very flexible with customization.
A middle ground between survival and creative servers, this type of server requires players to gather certain materials in the traditional way, but allows them to use infinite amounts of other materials, usually common materials like wood, stone, sand, and other plentiful resources.
This type of server is usually unmodified or very lightly modified with proprietary software. Subtypes of this server type including Anarchy and Chaos servers, Hardcore PvP servers generally have few rules and encourage killing, looting, and griefing other players.
A server in which players choose from a kit or several kits to have big PVP battles. The gamemode is either Adventure or Survival.
These servers are made up of various minigames, such the Survival Games and Spleef, and are usually heavily modified.
Constructed to emulate life in Prison, this type of server is comprised of a prison where players spawn, sometimes with multiple "cell blocks" which players advance through in order to gain freedom. This type of server must be modified with proprietary software to control players and simulate a Prison experience.
This type of server typically falls either into the Adventure or Survival categories, with the exception that Player-versus-Player (PvP) combat is allowed or encouraged in tandem with Player-versus-Environment (PvE) combat. Griefing is usually not allowed.
This type of server encourages or requires players to assume the role of their character. Usually this means that there is a system in place which allows the player's ingame character to be customized, frequently by assuming species, social ranks, superhuman abilities, and the like. These servers are frequently modified with proprietary software to provide players with special powers in exchange for choosing a player class, species, etc.
This type of server forces players to show their legitimate side. This requires proprietary software. Players will spawn on a small island with a chest full of basic items. They will be forced to survive with few items by creating a farm, a cobblestone generator, and other useful renewable devices.
The most popular current server type, Survival servers are typically played entirely in the Survival game mode. Maps on these servers usually consist of a premade spawn location, which is sometimes built by the server operators using creative mode, and a surrounding area created by players. These types of servers are frequently modified to suit the desires of the operators, and frequently have some kind of system which allows players to protect their structures from griefing, either on their own or with the assistance of an operator.