Persistent connections are links that do not close when the execution of your script ends. When a persistent connection is requested, PHP checks if there's already an identical persistent connection (that remained open from earlier) - and if it exists, it uses it. If it does not exist, it creates the link. An 'identical' connection is a connection that was opened to the same host, with the same username and the same password (where applicable).
People who aren't thoroughly familiar with the way web servers work and distribute the load may mistake persistent connects for what they're not. In particular, they do not give you an ability to open 'user sessions' on the same link, they do not give you an ability to build up a transaction efficiently, and they don't do a whole lot of other things. In fact, to be extremely clear about the subject, persistent connections don't give you any functionality that wasn't possible with their non-persistent brothers.
This has to do with the way web servers work. There are three ways in which your web server can utilize PHP to generate web pages.
The first method is to use PHP as a CGI "wrapper". When run this way, an instance of the PHP interpreter is created and destroyed for every page request (for a PHP page) to your web server. Because it is destroyed after every request, any resources that it acquires (such as a link to an SQL database server) are closed when it is destroyed. In this case, you do not gain anything from trying to use persistent connections -- they simply don't persist.
The second, and most popular, method is to run PHP as a module in a multiprocess web server, which currently only includes Apache. A multiprocess server typically has one process (the parent) which coordinates a set of processes (its children) who actually do the work of serving up web pages. When a request comes in from a client, it is handed off to one of the children that is not already serving another client. This means that when the same client makes a second request to the server, it may be served by a different child process than the first time. When opening a persistent connection, every following page requesting SQL services can reuse the same established connection to the SQL server.
The last method is to use PHP as a plug-in for a multithreaded web server. Currently PHP has support for WSAPI, and NSAPI (on Windows), which all allow PHP to be used as a plug-in on multithreaded servers like Netscape FastTrack (iPlanet), Microsoft's Internet Information Server (IIS), and O'Reilly's WebSite Pro. The behavior is essentially the same as for the multiprocess model described before.
If persistent connections don't have any added functionality, what are they good for?
The answer here is extremely simple -- efficiency. Persistent connections are good if the overhead to create a link to your SQL server is high. Whether or not this overhead is really high depends on many factors. Like, what kind of database it is, whether or not it sits on the same computer on which your web server sits, how loaded the machine the SQL server sits on is and so forth. The bottom line is that if that connection overhead is high, persistent connections help you considerably. They cause the child process to simply connect only once for its entire lifespan, instead of every time it processes a page that requires connecting to the SQL server. This means that for every child that opened a persistent connection will have its own open persistent connection to the server. For example, if you had 20 different child processes that ran a script that made a persistent connection to your SQL server, you'd have 20 different connections to the SQL server, one from each child.
Note, however, that this can have some drawbacks if you are using a database with connection limits that are exceeded by persistent child connections. If your database has a limit of 16 simultaneous connections, and in the course of a busy server session, 17 child threads attempt to connect, one will not be able to. If there are bugs in your scripts which do not allow the connections to shut down (such as infinite loops), the database with only 16 connections may be rapidly swamped. Check your database documentation for information on handling abandoned or idle connections.
There are a couple of additional caveats to keep in mind when using persistent connections. One is that when using table locking on a persistent connection, if the script for whatever reason cannot release the lock, then subsequent scripts using the same connection will block indefinitely and may require that you either restart the httpd server or the database server. Another is that when using transactions, a transaction block will also carry over to the next script which uses that connection if script execution ends before the transaction block does. In either case, you can use register_shutdown_function() to register a simple cleanup function to unlock your tables or roll back your transactions. Better yet, avoid the problem entirely by not using persistent connections in scripts which use table locks or transactions (you can still use them elsewhere).
An important summary. Persistent connections were designed to have one-to-one mapping to regular connections. That means that you should always be able to replace persistent connections with non-persistent connections, and it won't change the way your script behaves. It may (and probably will) change the efficiency of the script, but not its behavior!