In a light aircraft like a C172 or PA-28, during a climb the ball in the turn coordinator moves to the right. Why?
There are 4 left turning tendencies in aircraft with "clockwise" (as seen from the pilots seat) turning propeller, 3 of which may apply during a full power climb out. The four are:
Prop torque - prop turns right, plane rolls left - more noticeable at high power/slower speed
Gyroscopic precession - nose goes down - plane yaws left - not a factor in a climb - some effect as plane levels out, also there on tail draggers as rear of plane rises during take off roll
Asymmetric prop blade thrust at higher AOA (p factor). Differences in thrust of ascending and descending prop blades will cause yaw to the left.
Spiral slipstream striking vertical stabilizer and side of fuselage, causing yaw to left.
Number 1 has to be watched in higher performance engines during takeoff roll. Lifting off with insufficient airspeed will cause the plane to roll over and crash. Fortunately, in a Cessna 172, this tendency is much more modest but should not be ignored.
3 and 4 are probably the main culprits of left yawing tendency in the climb. The answer: pitch to safe climbing airspeed (generally Vy) at full power and "step on that ball" with right rudder.
Post script: the rolling tendency caused by high power lower speed torque may cause the pilot to feed in more right aileron when the better input is coordinated with rudder. As the plane accelerates from rotation and liftoff speed to Vy, it would be easy to miss you've rolled slightly to the right due to increased aerodynamic force. In straight line flight this would not be a major problem (unless you were dangerously near aileron reversal stall speed). I would work on glancing down at the ball as soon as safe climb is established, perhaps while checking airspeed. Stay coordinated.
If the ball is offset it means the airplane is accelerating laterally, that is, it's slewing to the side as it flies. Torque from the engine, plus the spiral flow from the propeller wake and asymmetric thrust from the propeller disc, is doing it and the higher the power setting/the lower the speed, the worse it is.
Vertical fins typically have a bit of directional offset to correct for this in the cruise case without having to add rudder (look carefully at the dorsal fin on a 172, and you'll see it isn't aligned perfectly down the centerline but points to the left a couple degrees), but at high power settings and low speeds, like in a climb, the fin offset isn't enough and some right rudder input is also required to center the ball.
It's the same reason you need a lot of right rudder on takeoff. If the engine rotated the other way, everything would be opposite and you'd need left rudder. Engines themselves are also offset in the mount, canted to the right a little bit, for the same purpose.
If you are sufficiently sensitive to it you will feel the side slip in your body, which will want to slouch in the same direction as the skid ball (because that's where the apparent gravity is) and you can maintain reasonably coordinated flight without looking at the ball.