Think about how you fly without an autopilot first. You establish a pitch attitude and power setting for the desired climb, cruise, or descent and then use elevator trim to eliminate your need to hold pressure (back or forward) on the elevator. You use the ailerons to steer the airplane, either to keep it straight or to turn as desired. And you use the rudder to coordinate your use of ailerons and counteract the various left-turning tendencies (and to a much smaller degree the occasional right-turning tendency). All airplanes (until someone shows me an exception) have elevator trim. Many small airplanes have some form of rudder trim so you at least do not have to maintain rudder pressure in straight and level cruise flight, although the rudder trim can be as simple as an adjustable spring. Many small airplanes have an aileron trim tab that can be adjusted on the ground and some probably have some form of aileron trim adjustable in flight, which in either case has the purpose of letting you use minimum aileron input during straight and level cruise flight.
So your actual flying consists of setting the airplane up with power and elevator trim to climb, descend, or cruise at a desired altitude and speed without you having to touch the elevator, using the rudder trim so at least in straight and level cruise flight you do not have to touch the rudder pedals to keep the ball centered, and using the ailerons to keep the plane on course. Now think about how to reduce your workload as a pilot with a minimum of automation.
Of the three controls, ailerons require the most pilot work because the plane has much less stability in the roll axis than in pitch and it's more sensitive to small excursions in steering than in altitude. (Go 10 feet high and you barely care. Go 10 degrees left of course and you'll end up at the wrong destination.) They get the first axis of autopilot control as a result. (As a bonus, an aileron-axis autopilot needs fewer data inputs than an elevator-axis autopilot does, so it is less expensive to make and install.) And for many pilots, that's enough, because in a small plane the elevator trim and rudder trim make it easy to get the plane flying level and coordinated at cruise without the pilot having to touch the controls. A single-axis autopilot can be as simple as a wing leveler or as complex as a GPS-driven flight management system that flies your entire route for you.
Elevator is the second axis to get automated because, be honest, do you really want to constantly make tiny trim adjustments or tiny elevator inputs just to deal with 10-foot excursions from your desired altitude or small deviations from your desired climb or descent airspeed? And most two-axis autopilots I've seen either tell you to trim up or down so they don't have to work as hard (just as you would do if you were doing the flying) or engage electric trim to get rid of control forces on their own. They basically fly the airplane the same as you do and can vary from simple altitude hold to complicated systems that let you pre-select an altitude and then have the autopilot control pitch (and, on big enough planes, power) to maintain a vertical speed, an airspeed, or other parameter for the desired climb profile, possibly even making in-flight altitude changes according to your flight plan if they are fancy enough.
Rudder is the last axis to be automated, because in cruise flight it hardly enters in to the equation unless you're making significant turns, and most simple autopilots don't like to make significant turns anyhow. In small planes, you as a pilot should coordinate every little aileron input with proper rudder input, but in reality the plane will not enter a cross-control spin just because you didn't coordinate a half-standard-rate turn, so the autopilot can also get away with that kind of lazy flying and nobody really complains. And when the rudder does enter into the equation, you use it to keep the ball centered, so that's what a yaw damper does. Whether in an extended high-power high-AOA climb, a Dutch roll, or a steep turn, the yaw damper applies rudder pressure to center the ball for you (or for the rest of the autopilot system). And that's an independent action from what the rest of the pilot or autopilot is doing, so it ends up being a separate system as others have described above.
