A manual transmission is connected to the engine through the clutch, which consists (among many other things, but most importantly) of a flywheel, driven by the engine, a clutch plate which drives the transmission when connected to the flywheel, and a release fork, which is connected to the clutch pedal. When you press the clutch pedal, a cable or hydraulic piston pushes the release fork, pulling the clutch plate away from the flywheel and disengaging the transmission from the engine. More about this later.

The transmission on a 5-speed consists of six different gear ratios (5 forward, one reverse). Here are some typical gear ratios:
1st: 2.315:1
2nd: 1.568:1
3rd: 1.195:1
4th: 1.000:1
5th: 0.915:1

A shaft attached to the clutch plate enters the transmission. Whenever the clutch is engaged, if the engine is on, this shaft is turning. This shaft ends in a gear wheel that is meshed with the gear wheel at the end of the layshaft (the gear ratio here is 1:1). The layshaft is one long shaft that contains all the gears. Whenever the driveshaft is turning, the layshaft is turning, and all the gears are turning.

Above the layshaft is a splined shaft that connects directly to the driveshaft through the differential to drive the wheels of the car. If the wheels are spinning, the splined shaft is spinning. On the splined shaft are gear wheels meshed with each of the gear wheels on the layshaft. The gear wheels on the splined shaft are mounted on bearings, so the splined shaft does not necessarily spin when they spin. These gear wheels are always meshed with the gear wheels on the layshaft. This means that if the engine is on and the clutch is engaged, all six gears are turning. However, since they are mounted on bearings, only the gear that is engaged is turning the wheels. If the car is in neutral, the splined shaft (and the wheels) can spin independently of the rest of the engine and transmission. Mounted on the splined shaft between every other pair of gears is a collar. Most gear shift knobs these days look like this:

1 3 5
| | |
2 4 R 

So, there is a collar between 1 and 2, 3 and 4, 5 and R. Attached to each collar is a shift rod, attached to the gear selector fork, attached to the gear shift knob (i.e., the stick). The shift rod moves the collar between the two gears it is associated with. Moving the stick from neutral to the left chooses the shift rod that moves the collar between gears 1 and 2. Moving the stick forward from there moves the collar into gear 1. The collar is a small cog with dog teeth; each of the gears on the splined shaft (which, remember, are usually mounted on bearings) has a hole in the middle that the collar fits into. When the collar is in a gear, the clutch is engaged, and the engine is on, the wheels turn. Modern gearboxes have a synchromesh system whereby a conical protrusion from the collar with a high coefficient of friction (and a similar indentation in the gear) allows friction to bring the gear to the right rpm rather than using double clutching. A quick note on how reverse works: between the reverse gear on the splined shaft and the corresponding gear on the layshaft is a very small idler gear. This means that the reverse gear on the splined shaft is always spinning in the other direction from every other gear. This also means you could never throw the car into reverse until you come to a full stop (or pretty close). You could, however, with enough force, destroy a collar.

Starting a manual
Remember, if the clutch is engaged and the car is in gear, when the engine is on the wheels turn. Therefore, it is suggested that you either start your car in neutral, or in 1st with the clutch down (modern cars wont let you start in any gear, neutral or otherwise, without the clutch down, so you might as well use the second method). Now, remember the clutch pad and the flywheel? The flywheel turns the clutch pad only through friction. This is important, because when the clutch is fully engaged and your foot is off the accelerator (and not on the brake), that will be the lowest speed in that gear - with the clutch fully engaged. To go slower (i.e., to start moving from a standing stop - no matter how fast you want to pull away, at some point between 0 and 10mph you'll still be doing 1mph) you must induce slippage by having the clutch pad half-pressed against the flywheel. This is called the biting point in Britain and the catch point in the States.

So, you are now in first gear, clutch pedal fully depressed, engine on. If the handbrake is not already up, pull it up. Press lightly on the accelerator (how hard you press depends on how quickly you want to pull away). Slowly release the clutch pedal, listening to the engine. At some point, the sound will change, and you may feel the car straining forward. This is the biting point. Release the handbrake, leave the clutch where it is, keep pressing the accelerator: you will slowly move forward. When you have started to move, release the clutch pedal (leaving it at biting point too long will wear your clutch pad out very quickly, and thats a $300 repair waiting to happen), and accelerate. If you release the clutch pedal too early, the car will probably move forward in lurches and stops before it finally stalls, because the engine is unable to complete its cycle (too much inertia in the wheels; wheels can't accelerate as quickly as the engine wants them to; no slippage in the clutch because youve engaged it completely).

Driving a manual
When you hear the engine begin to whine, its time to change gears. Push the clutch pedal, move out of one gear, move up a gear, release the clutch pedal, all in one fluid motion. The reason you have to use the clutch pedal is because with the clutch engaged, the engine is exerting force on the entire apparatus, including the collar. If you try to move the collar out of the gear while force is being exerted on it, you can damage it pretty easily. As you move into gear, the synchromesh system synchronizes the speed on the collar and the gear, saving you the trouble of double clutching, which was necessary in older cars.

When stopping (at a light for example), brake as normal, and towards the end push the clutch pedal. Once you are stopped, bring up the handbrake and shift into first; start as described above. If ever you feel the car begin to shudder, press the clutch pedal immediately: this means you are going too slow for the gear and below the engine's lowest rpm; downshift, and release the clutch.