i believe the main story with leaving a pc on/off comes from the same issue that is demonstrated with lightbulbs.
do you notice that a lightbulb almost always burns out right when it is turned on?
this is because of the initial surge of power that goes threw the filiment heating it up right away. while on, the filiment of a lightbulb will slowly deteriorate and form cracks. over time these cracks become bigger and bigger. when its old enough, the cracked up old filiment can't handle the initial great surge of power that flashes threw it right when it is turned on and breaks. it was tested by mythbusters that a lightbulb will use 3-5 times more power in the few milliseconds that its being turned on then while its running.
the story is that this initial surge of power can also harm computer circuitry right when turned on which is why many power supplies still supply power to motherboards even when the computer is turned off. wether or not this was actually tested i don't know. on the otherhand, as some have mentioned, leaving it on *could* be just as bad. if something goes wrong and the computer overheats, something will break. or if the computer doesn't have adequet cooling something could easily overheat if constantly left running.
but, as many have said, i have noticed no difference in any of my computers health when being turned on/off or being left running 24/7.
only thing i notice is the heat in my room from the computer

if i turn off my gaming pc my room will stay cooler. but if i leave it on my room warms up. but even this isn't the computer itself, its the monitor. a CRT monitor will put out alot of heat. if i have both my server pc's monitor and gaming pc's monitor on in my room for a day it will heat up to over 80F when the rest of the house is only around 60. but with just the gaming PCs monitor on my room will only get upto around 70F.