What is the difference between a torque wrench and a regular ratchet that you'd normally use with sockets? When should you use one vs. the other in automotive applications?
Torque wrenches are used for adjusting the specific tightness of nuts and bolts. They come with an adjustable torque setting which is usually in foot pounds (english) or Newton meters (metric). In automotive applications, most nuts/bolts will have a torque setting specified in the manual that when applied will give the exact amount of pressure needed to operate smoothly.
Regular ratchets have no adjustable torque setting. Typically you would refer to the tightness of what you were working on as 'hand-tight'.
For lugs nuts on wheels for example; you would definitely want to use a torque wrench to ensure the nuts are not too loose or too tight. By using a torque wrench, you can ensure that all of the nuts are tightened with the right amount pressure, which would more than likely be imperfect when using a regular ratchet.
A torque wrench only allows you to tighten a bolt to a torque spec, which you can adjust on the wrench. These torque figures are often quoted in shop manuals/specs (eg. "tighten valve cover bolts to 30 ft-lbs") because many bolts were designed for a specific amount of torque to work properly. For example, if you over-tighten a bolt on a gasket, you could cause it to fail due to the extra pressure. You should use a torque wrench as specified by the shop manual for a given repair, especially if the bolt seems important - you don't really need a torque wrench for bodywork bolts etc., but you definitely need one for most bolts you'll find while taking apart an engine.
As the other answers have said, the torque wrench (TW, to save typing) allows you to tighten a nut/bolt to a specified torque, either by 'clicking' off or by showing a values with a needle (old fashioned needle type ones).
Never use a TW to undo anything, as you'll damage it. I personally only use the TW for the final tightening, using a normal ratchet for everything else - wind the nut/bolt on using a ratchet until it's nearly on, then switch to the TW to finish it.
Always store the TW unwound (i.e. set to the lowest torque setting), otherwise the spring will start to lose it's springyness.
A socket wrench is like a small steel tube - one end has ridges that fit precisely around a bolt. The other end often has an opening of 1/4", 3/8", or 1/2" square. This end receives the protruded part of a ratchet with the same dimension.
Most torque wrenches look like a larger version of a ratchet -- in fact, it often also has a ratchet mechanism.
The purpose of a torque wrench is to measure the precise amount of pressure you use to tighten down a bolt. Generally, you do this by placing a Socket wrench for the bolt size in question onto a Torque wrench. You will then dial (usually on the handle end) to the amount of torque the specification calls for -- usually in Foot lbs or Inch lbs.
Affix the bolt receiving end to the bolt and tighten, clockwise. You will hear a clicking sound when you have reached the indicated torque.
Again, a Torque wrench is used to measure the precise amount of pressure(torque) applied to a bolt. This is usually done to assure you do not over-tighten the bolt and thereby strip threads AND/OR assure you do not under-tighten and the bolt gets loose while you are operating the bike(a rather dangerous situation).
A torque wrench can cost over $50.00, and you will likely rarely use it. You might just tighten the noted nuts/bolts firmly, then take it to a local bike shop and ask if it is tightened enough. They might take a torque wench to it to check it out. However, experienced assemblers generally just firmly tighten --unless it's a high-end bike.
There are less expensive -- but less precise --torque wrenches available. You would still affix a socket wrench on it. Difference: instead of a dial adjuster on the handle with the audible click, it has a floating pointer that points to the measurement imprinted at the head of this wrench.
If you do decide to buy a torque wrench, they are available at Sears, Home Depot, and probably at most large auto parts stores(Pep Boys, Kragen, etc). Some of these auto parts stores might even allow you to rent one. You can also buy the needed socket wrench at these same stores.