SQL is a declarative language that allows you to create and manipulate objects, as well as data. You can group SQL statements into scripts so that you can run the scripts in a more predictable and reproducible way. However, such scripts are seen by PostgreSQL as a sequence of unrelated commands, that is, you are responsible for correlating such commands into appropriate scripts. Things get even worse when you have to deal with foreign languages or binary libraries; the cluster knows nothing about your aim and how every single object is related to each other. Luckily, extensions help in getting order out of chaos.
An extension is a packaged set of files that can be installed into the cluster in order to provide more functionalities, therefore to "extend" the current cluster set of features.
An extension can be something general, like a new data type, a new index type, or a service to send emails directly from within PostgreSQL, or it can be something really...
