A database is a named collection of SQL objects ("database objects"). Generally, every database object (tables, functions, etc.) belongs to one and only one database. (But there are a few system catalogs, for example pg_database, that belong to a whole cluster and are accessible from each database within the cluster.) More accurately, a database is a collection of schemas and the schemas contain the tables, functions, etc. So the full hierarchy is: server, database, schema, table (or some other kind of object, such as a function).
When connecting to the database server, a client must specify in its connection request the name of the database it wants to connect to. It is not possible to access more than one database per connection. (But an application is not restricted in the number of connections it opens to the same or other databases.) Databases are physically separated and access control is managed at the connection level. If one PostgreSQL server instance is to house projects or users that should be separate and for the most part unaware of each other, it is therefore recommendable to put them into separate databases. If the projects or users are interrelated and should be able to use each other's resources they should be put in the same database, but possibly into separate schemas. Schemas are a purely logical structure and who can access what is managed by the privilege system. More information about managing schemas is in Section 5.7.
Databases are created with the CREATE DATABASE command (see Section 19.2) and destroyed with the DROP DATABASE command (see Section 19.5). To determine the set of existing databases, examine the pg_database system catalog, for example
SELECT datname FROM pg_database;
The psql program's \l meta-command and -l command-line option are also useful for listing the existing databases.
Note: The SQL standard calls databases "catalogs", but there is no difference in practice.