PSP (Paintball Sports Promotions) relates to a tournament series, and PSP mode is the mode of fire prescribed by the rules of that tournament series.
PSP mode is an enhanced (ramping) mode, that allows your marker to fire quicker than you are actually pulling the trigger. The marker will shoot in semi auto until the enhanced mode is activated, which occurs when certain conditions are met.
In the case of PSP mode, originally, three shots had to be fired in semi auto at a rate of 5bps or higher. If this rate of trigger pulls was then maintained, the marker would then fire at a maximum allowed rate if 15.4bps, though most used the extra 0.4bps as a safety buffer, and set their markers/boards to 15bps.
Different versions of the PSP mode were devised to enhance the rate of fire in different ways, but the official PSP mode was based on the use of 3 round bursts on each trigger pull to enhance the rate of fire.
PSP mode has a one second buffer, meaning that your trigger pull rate can drop below 5pps for a second or less, or you can even stop pulling the trigger for a second or less, but as long as you then start pulling the trigger at the specified rate, the marker will still ramp. Start pulling the trigger *after* a second or more has elapsed, and the marker will revert back to semi until the ramping is reactivated.
Recently, the rate of fire setting for PSP mode has been reduced to 13.3bps maximum.
Millennium mode is a version of ramping used in some European Tournament series. Originally, it was very similar to PSP ramping, with a 5bps activation rate and 15bps maximum rate of fire. The only major difference was that there was no one second buffer for ramping. If you stopped pulling the trigger, the board switched back to semi; if your trigger pull rate dropped below 5pps, the board switched back to semi.
The Millennium committee than changed the prescribed mode for their tournaments, upping the trigger pull rate to activate ramping to 7.5pps. Maximum rate of fire stayed at 15bps, and there was still no buffer for ramping.
Most recently, for the 2008 season, the Millennium mode has changed again. The maximum rate of fire is now 12bps, and the ramp activation rate is the equivalent of 6pps after three shots in semi auto.