The function of any transmission is transferring engine power to the driveshaft and rear wheels (or axle halfshafts and front wheels in a front-wheel-drive vehicle). Gears inside the transmission change the vehicle's drive-wheel speed and torque in relation to engine speed and torque.
Regarding this, what is the difference between an automatic and a stick shift?
The primary difference between an automatic and a manual transmission is the method the car uses to shift gears. In an automatic transmission, the car decides when you shift and automatically changes gears for you. In a manual transmission car, you are responsible for shifting gears using the clutch and gas pedals.
Are manual and stick the same thing?
Nothing, it's two terms for the same thing. I'm not 100% sure where they use the term standard shift, but stick shift is used in the USA, but manual shift is used in the UK. Ultimately, they all refer to a car where the driver changes the gears, as opposed to an automatic, where the car does it for you.