Sync is everything
There's no way I can answer /all/ that right here, but here's a basic guide:
There are two types of sync. Real-time, and Tempo-based.
First the real time:
Realtime sync carries absolute time-reference from one thing to another. An example of a real time position is 01:34:20:02, which is expressed in hours, minutes, seconds, frames.
SMPTE LTC (linear time code?) is the audio form of real-time sync. That is, SMPTE travels via audio equipment. You can record SMPTE on tape, and you can cable it just like any other audio signal. You can even hear it - sounds like a turkey running around or something.
MTC is MIDI Time Code. Its basically the same thing as above, only over MIDI cables. This is a popular way of getting SMPTE LTC into computers (you just need a PPS2 or similar to convert the media).
VITC, Vertical Interlaced Time Code, is real-time sync over video. I don't know much about this.
Now for the Tempo-based:
Tempo-based sync is used mainly for syncing MIDI to Audio. An example of a tempo-based location would be something like Bar 10,
beat 4, quarter-note-part 22. In order to sync with a real-time device, you need either a computer, or an MPC studio-edition (I think), which has been made aware of the song's start time, and tempo.
FSK, Frequency-Shift Keying. This isn't used much anymore, mainly because of how popular computers in studios are. But here's what it is. Its tempo-based sync over audio. If you had a MIDI sequencer in the studio, and you wanted it to play back with your tape, in sync, you just stripe FSK down on the tape, convert it to MIDI (with a PPS2 or similar), and you're done. You can record it on tape, and move it around your studio with regular audio cables. Sounds a lot like SMPTE.
Midi Clock is the MIDI form of Tempo-based sync. Lots of devices accept and understand Midi clock.
The thing is though, a computer provides so much flexibility in changing real-time sync into tempo-based sync, that these tempo-based syncs are rarely used alone now. Studios without a computer, however, probably use them frequently.
I guess we might as well go over actual studio clocks now, as that was the topic of your post.
As DDev said, studio clocks are used to generate a common 'pulse' for digital gear. Imagine this, you have a DAT player with AES/EBU out, connected into a cd recorder, with AES/EBU. If the two clocks are off slightly (and at 44,100 samples per second, you can bet they will be a little different), data will get sent, when the other device isn't looking for it. And then you get these drop outs, and similar. Its very bad sounding

.
So what people will do, is either patch "word clock" between the devices to keep everything in sync (and set one device to 'slave' to the other, usually 'ext' for external), or use the timing within the incoming digital signal.
This works great, but then you have situations with so much gear, it would be impossible to sync them all up to one dat recorder, due to the signal loss of 'splitting' word clock 20 times or so. The average digital studio has so much gear, it would be a nightmare.
So, they have these things called studio clocks, which provide stable, distributed clock pulses to everything. The better the clock source, the lower the jitter (variations in sample rate, which I think would translate into unstable pitch on playback). Seemingly bad converters might sound half way decent with a better clock source.
There's also machine sync, which is used to control the location and transport of two or more machines. For example, you want the dat recorder to start recording the moment you hit 'play' in protools. This can be either over midi, (MMC - Midi Machine Control), 9-pin serial, or something completely strange, like the 15-pin connector on the back of an Otari tape machine.
Sync is really important to understand, in my opinion, because it opens up possibilities that would otherwise be, well, impossible.
