PostgreSQL consists of a set of server processes, the group leader of which is named the postmaster. Starting the server is the act of creating these processes, and stopping the server is the act of terminating those processes.
Each postmaster listens for client connection requests on a defined port number. Multiple concurrently running postmasters cannot share that port number. The port number is often used to uniquely identify a particular postmaster and hence also the database server that it leads.
When we start a database server, we refer to a data directory, which contains the heart and soul—or at least the data—of our database. Subsidiary tablespaces may contain some data outside the main data directory, so the data directory is just the main central location, and not the only place where data for that database server is held. Each running server has one data directory, and one data directory can have, at the most, one running server (or instance).
To perform any action for a database server, we must know the data directory for that server. The basic actions we can perform on the database server are starting and stopping. We can also perform a restart, though that is just a stop followed by a start. In addition, we can reload the server, which means we can reread the server's configuration files.
We should also mention a few other points.
The default port number for PostgreSQL is 5432. That has been registered with the Internet Assigned Numbers Authority (IANA), and so it should already be reserved for PostgreSQL's use in most places. Because each PostgreSQL server requires a distinct port number, the normal convention is to use subsequent numbers for any additional server; for example, 5433, 5434, and so on. Subsequent port numbers might not be as easily recognized by the network infrastructure, which might, in some cases, make life more difficult for you in large enterprises, especially in more security-conscious ones.
Port number 6432 has been registered with IANA for PgBouncer, the connection pooler that we will describe in the Setting up a connection pool recipe. This happened only recently, and many installations are using nonstandard port numbers such as 6543 only because they were deployed earlier.
A database server is also sometimes referred to as a database cluster. I don't recommend this term for normal usage because it makes people think about multiple nodes, not one database server on one system.