Manual Transmissions
A manual transmission (or manual transaxle on transversely-mounted applications like a front wheel drive vehicle) requires driver interaction to change gears, which also requires use of a clutch (with the exception of special situations). Because an engine has a limited range of RPM it can operate within (stalls out when too low, parts are working too hard when too high), a transmission is required to give a proper gear ratio for the engine to work with at any given speed. The gears also allow for the generation of extra torque at lower vehicle speeds thanks to mechanical advantage offered by low speed gears.
Gear Count
A transmission is often titled a "(number)-gear transmission", and that number isn't a count of actual gears within the transmission (there are usually more than double that amount). That title tells you the number of forward speeds it has. Every "speed" is two physically connected gears with a ratio that differs from the other gears.
Gear Ratios
A gear ratio is a comparison of the number of teeth on the driven (output) gear divided by the drive (input) gear's tooth count. An example of this would be that a gear ratio of 3:1 (said "three to one") and would have gears with a number of teeth like 27:9, 30:10, and so on. In that example, the driven gear will turn once for every three turns of the drive gear. The output speed will be three times slower, but it will provide three times the torque.
Gear Selection
All of the forward gears (the reverse is operated differently) in a manual transmission are meshed at the same time. In order to select a gear to use, a gear must be meshed to the output shaft. This is done by sliding a collar that is constantly meshed with the output shaft to a desired gear. The gear has "dog teeth" that the sleeve can also mesh with, and in most modern transmissions, a synchronizer sleeve helps match the speed of the selector sleeve with the speed of the gear. If the speed isn't matched, you will "grind gears" which can cause damage to the dog teeth.
Manual Transmission Lubricants
There are many types or lubricants used in manual transmissions. Some of them use lubricants designed for their specific demands of that transmission model, while others just take motor oil or even automatic transmission fluid. Historically, gear oil was used in most, but it is a thick, viscous fluid, so more recent transmissions have seen a move to lower viscocities for improved fuel efficiency.
Gaskets & Seals
Manual transmissions sometimes have an oil pan, which will have a gasket to seal the mating surfaces. There will also be an input shaft seal and an output shaft seal, or two output shaft seals on a manual transaxle.