Instance is the term used for an area in the game that is created separately each time a player, or Group, enters that area. As opposed to the rest of the world/game which exists in common for all players. Instances are used to create areas where the developers feel it is important to keeping players/groups from interacting with other players while in that area. The most common / well-known use of an instance is for Dungeons.
In some games every mission takes place in a unique instance, in others only Dungeons are instanced. The Secret World is in the middle of these extremes. In addition to Dungeon instances there are some missions that use instances. A small number compared to the total number of missions available, but players are sure to encounter several as they progress through the game. Missions that use instances may take place entirely within an instance, only have certain tiers within an instance, or have various tiers in various instances.
Additionally, instances can be either solo-only or opened to groups. For solo-only instances - players can be in a Group and do not have to drop out of the group to enter. If multiple players that are grouped together enter the instance (typically because they are all on the same tier of a mission they are running together) then they will all be placed into an instance of that area that created uniquely for them and will not be able to see or interact with other group members (other than via chat). Dungeons will strongly recommend that you have a full group of 5 when you enter, if you don't, but do not require it and it is quite possible to enter a Dungeon solo. While there are missions that require you to achieve objectives in a solo-only instance, there are no missions that require you to enter as a group.
For a list of Instances, see Category:Instance locations