It's a connection that lasts beyond the lifetime of the script's execution; roughly, if you use [DB]_pconnect, you're telling Apache (via the PHP module) to keep the connection up after the script exits. When the next script calls pconnect, PHP will check to see whether there are any open, and if so, it will assign your db handle to the open connection.
Note: you must be using PHP as a module, not a CGI.
If there's considerable overhead in establishing connections to the DB (which, I believe Tim said, is a significant issue with Postgres), it makes sense to use persistent connections. Even if you just find the wait to get the script annoying, you might want to look into using them.
HTH,
AC