Let's just get this out of the way first: if it doesn't have three pedals and require both hands to drive, it's simply not a car. It's a sad, sad excuse for transportation.
A manual transmission lets the driver select which gear they would like to be in, and control how they shift. By depressing the clutch (operated by one's left foot), the engine is disconnected from the wheels. One can now use the gear shift lever to select an appropriate gear. The gear shift lever typically operates a set of selector forks in the transmission. These forks engage different gears, depending on the position of the gear shift lever.
One a gear is selected, one slowly lets the clutch out while increasing the amount of gas given to the engine. The clutch is essentially a plate connected to the engine and a plate connected to the wheels that mesh together. When the clutch pedal is to the floor, the plates are completely separated. By letting the clutch out, one slowly brings the plates together. It is incredibly important to let the clutch out slowly: when the clutch is in, the plates are likely rotating at different speeds. By bringing them into contact slowly, one allows slippage between them before full engagement takes over, which means that they can accelerate each other until they are rotating at the same speed.
Increasing the amount of gas given to the engine while letting the clutch out makes one less likely to stall. Stalling occurs when the engine isn't producing enough power to continue to rotate, and so it simply seizes. This is deeply embarrassing, especially when it occurs at a busy red light (ask me how I know...). In letting the clutch out one is engaging a gear, and therefore increasing the load on the engine. Pushing the gas pedal in more lets the engine produce the power the transmission is now calling for.