Persistent connections are SQL 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 SQL link, they do not give you an ability to build up a transaction efficently, 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.
Why?
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 each 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 serviced by a different child process than the first time. What a persistent connection does for you in this case it make it so each child process only needs to connect to your SQL server the first time that it serves a page that makes us of such a connection. When another page then requires a connection to the SQL server, it can reuse the connection that child established earlier.
The last method is to use PHP as a plug-in for a multithreaded web server. Currently this is only theoretical -- PHP does not yet work as a plug-in for any multithreaded web serve