A massively multiplayer online game is one that is played by a large number of people simultaneously in a persistent game world. Common types of MMO's are Role playing games (MMORPG), First person shooters (MMOFPS) an Real time strategy games (MMORTS).
Mechanics
Grouping
Players can form a group and explore and fight together. As a group the share rewards such as XP and loot and the can chat and see each other on the world map. A member of the party is selected the leader and can change the settings for the group and invite and kick members of it.
Raid
A raid is like a larger group to take on some major challenges.
Guild
Members of a guild can share resources and help each other.
Instance
A instance is when you get a private version of a dungeon for you or your group. It makes it possible to complete an area without other players running around and messing things up.
World bosses
Public dungeons
Public raids
Group quests
Phasing
With phasing each player can see a different version of the same area. A NPC can exist for one player and be gone for another player. One player can see a empty plane and another a castle. For the major changes the players need to only see other players in the same phase as them.
Housing
This let the player buy a house in the world. There are often a level limit and costs to buy and rent the house. The house and the neighborhood of the house is often instanced to be able to have any number of player houses. The house can then often be decorated with furniture and other items by the player. Another option is guild housing where guilds can buy a guild hall.
PvE
In PvE the players fight against the monsters and challenges in the game world.
PvP
In PvP the players fight against other players. RvR
Trading
Rmt, Trading, Auction
Chat
Realm, help, guild, lfg, trade
Economy
Server events
Transportation
In a MMO world it takes time to move between areas so some form of Rapid-Transit system is useful to avoid the game turning into a walking simulator. A Rapid-Transit system moves the player to the destination without any further input from the player.
Tech
Shards
It's often hard to keep all the players on the same server in a popular MMO so the world is often split into shards. Each shard is a separate instance of the whole world.
Games
The Elder Scrolls Online - 2014
Star Wars: The Old Republic - 2011
World of Warcraft - 2004
Ultima Online - 1997
Reference