Objectively speaking, PRaT is directly related to the speed of a system's response and is the continuously relative delay between the information being read from the source and the tones being reproduced from the driver. The design factors that will have perhaps the largest impact on this would be the quality of the power supply in the amplifiers and the speed of the materials used in the driver material.
There are lots of opinions about this out there, and if you google "flat earth audio" you will undoubtedly find a link to the flat earth forums where this has been debated into the ground.
Legend has it that the foot-tapping description was attributed to Linn, who encouraged their dealers to tap their feet when demoing Linn systems and not tap their foot with competitors equipment to demonstrate the rhythmical superiority.
Now, to add a bass player's perspective (adding to the drummer's) the PRaT of a system just means that large transient attacks of a bass are not softened by a slow amplifier and the punch of the instrument is properly presented to the listener. My current bass amp is small by most standards but even then a home hi-fi would have a considerable task to replicate the punch of a 400W dedicated amplifier and two 12" long voice coil drivers. Systems with PRaT do try, and some do it very well.
I've owned systems from a British manufacturer known for PRaT and found that it got the PRaT factor right in many ways, but the overall balance of the music gets lost in the process. Speed at all costs, if you will. I've moved on to a newer tube-based design from Audio Research that strikes a better balance.
In the end, this is simply another audiophile discussion point. There are a million and one ways to tell other people how your system is different than everything else on the market, and PRaT is popular with British audio fans to do just that.