Here's a link to an article that describes what you're looking for in detail.
What the gearbox does is allow the engine to run at the RPM range that it is most happy, while allowing your wheels to run at the speed which you desire. It does that through a collection of gears who's ratio determines how fast the wheels turn for a given engine speed.
All of the following comes from the article on transmissions.
The green shaft comes from the engine and spins at engine speed as long as the clutch allows the engine to be connected to it.
All the gears on the red shaft spin together.
All the blue gears on the yellow shaft are mounted on splined gears so they can shift back and forth.
The purple collar is attached to the gear shift knob, which changes which blue gear is attached to the red gears. This example is only a two speed transmission, but it's the same principle for any manual transmission.
The clutch is the device that connects and disconnects the engine from the rest of the transmission. This allows the gears to be changed without that wonderful grinding noise you mentioned. That grinding is cause by the gear ratio you have selected not matching the engine speed to the wheel speed very well. This happened a lot more before syncromesh which is a system for bringing the gears together gradually rather than slamming them together. The slamming required you to manually get the engine RPM correct with the gas pedal to match the engine RPM to car speed through the gear you selected. If you missed on the RPM, you got the crunch. Synchromesh brings the gears together gradually, allowing the two gears to find each others speed without you having to get it exactly right with the throttle.
Back to the clutch, here's a photo of one from another linked article
The clutch plate is the disk with the springs on it all the way to the left. The clutch plate is two plates that are connected together in such a way that they can slide over each other for a short distance. The springs keep the plates together, but when the engine is first connected to the gears (by letting out the clutch pedal) the springs compress to put some give in the system and prevent a lurching forward as the connection is suddenly made. The clutch sits between the engine and the green shaft in the first picture, allowing the driver to connect the engine to the wheels or not, via the clutch pedal.
Link to the article on clutches for more info:
I think that was all the parts of your question..
Hope that helps!