Thoroughbred and Palomino are different processor cores for the Athlon XP line. Thunderbird chips are their predecessors. Athlons from I believe 750MHz up to 1.4GHz, that are not Athlon XP, are T-Birds.
Now, a thoroughbred is a 0.13 micron CPU architecture, while Palomino is 0.18. There are also TWO Thoroughbred cores (A & B), just to keep the mix a little more confusing. The older Palomino's are all phased out now as far as I know, and all Athlon XP chips being manufactured now are Thoroughbred (all the newer ones are going to be the B core).
There is no compatibility difference between all these guys. The major difference lies in the fab size (0.13) and the power dissipation (the T-Breds run at a lower Vcc voltage than Palomino cores).
Depending on when you bought it, your XP1800+ is probably a Palomino chip. One way to tell is by looking at the actual CPU. Palomino's have the chip info etched onto the chip in the center, while T-Breds have that info on a black label that is stuck onto the green/brown material.
Hope this helps!

P.S. If you want to add another, the Athlon XP also has another core known as the Barton, which has a 512 cache on it instead of the 256!
P.P.S. Sorry, but I don't know the MP trick. Will stay tuned!
edit~booerns! Bob beat me!

Well the visual helps my post out a little.