A ramp agent at the airport assists the airplane coming into the gate (Most of them are aircraft mechanics). They also help push the airplane out to a certain distance from the gate (when the plane leaves for take off). Ramp agents also help with baggage loading and unloading.

An airplane parks on the part of the airport known as the "ramp" or "apron"

It depends on the airport--the terms they use are "parking fees" and "ramp fees."

Some airports have no fees. A moderate-size, not overly busy airport won't have ramp fees.

Some airports have fees that are waived if you buy a certain amount of fuel. And some have non-waivable fees.

Not usually, no.

The "ramp" or "apron" is the part of the airport where airplanes park to load or unload.

"Tarmac" is a more generic term referring to any paved part of the airport. It may be used to refer to the ramp, the taxiways, the runways, or indeed even when referring to the pavement itself in a more general sense.

Do you mean the long ramp that you walk down to get from the airport terminal onto the aeroplane which often slopes a little? If so, the correct name is a Jetway.