James,
Strictly; a "Live Centre" is the one in the Headstock Spindle, when you are turning "between centres" in an engineering or woodturning lathe. Watchmakers also turn between a pair of dead centres, and drive the work directly with a bow. [This is a simlar idea to the old "Bodger's" Pole Lathe.]
…. Several new terms to search for on Google, I suspect.
However: You are probably thinking of using either a dead centre or a rotating centre, in the tailstock. … In which case the big practical difference is that the dead centre needs to be kept lubricated; whereas the rotating centre should have a nice ballrace to take the load.
Hope that helps
MichaelG.
.
These are very good … and priced accordingly 
Edited By Michael Gilligan on 16/04/2013 23:17:11