A problem with many JDBC drivers is that only one thread can use a Connection at any one time -- otherwise a thread could send a query while another one is receiving results, and this would be a bad thing for the database engine.
The PostgreSQL JDBC Driver is thread safe. Consequently, if your application uses multiple threads then you do not have to worry about complex algorithms to ensure that only one uses the database at any time.
If a thread attempts to use the connection while another one is using it, it will wait until the other thread has finished its current operation. If it is a regular SQL statement, then the operation consists of sending the statement and retrieving any ResultSet (in full). If it is a Fastpath call (e.g., reading a block from a LargeObject) then it is the time to send and retrieve that block.
This is fine for applications and applets but can cause a performance problem with servlets. With servlets you can have a heavy load on the connection. If you have several threads performing queries then each but one will pause, which may not be what you are after.
To solve this, you would be advised to create a pool of connections. When ever a thread needs to use the database, it asks a manager class for a Connection. The manager hands a free connection to the thread and marks it as busy. If a free connection is not available, it opens one. Once the thread has finished with it, it returns it to the manager who can then either close it or add it to the pool. The manager would also check that the connection is still alive and remove it from the pool if it is dead.
So, with servlets, it is up to you to use either a single connection, or a pool. The plus side for a pool is that threads will not be hit by the bottle neck caused by a single network connection. The down side is that it increases the load on the server, as a backend process is created for each Connection. It is up to you and your applications requirements.