Tricycle vs taildragger
Is this required for all the planes (irrespective of size, type) to
land only on the back wheels?
All aircraft touch the runway on their main gear, and the main gear is at the level of the wing. Adding a wheel aft or forward of the main gear leads to two configurations with a noticeable difference: The angle between the wing and the horizontal plane when the aircraft is on the ground.
Those with a nose wheel and a low angle, the vast majority of today's aircraft, and all commercial airliners. This gear configuration is named tricycle.
Those with a tail wheel and a high angle, a category which was used at the beginning of aviation. This gear configuration is named taildragger.
The advantage of the tricycle is the stability on the ground, thanks to the engine being located between wheels. The inconvenient is that takeoff cannot take advantage of the weight of the engine to help rotate the aircraft. Elevators are more solicited. If the rotation is not executed correctly, a tailstrike can occur.
Tricycle aircraft landing
Does it have some advantage to it or is it a rule?
It is a technical requirement. When landing, the aircraft must touch the runway with a very small vertical speed, the glide path must slope gently down. The weight of the aircraft pulls it down at an accelerating rate (free fall), the aircraft must oppose this acceleration by generating lift.
Lift is proportional to airspeed and to the angle between the direction of the aircraft and the direction of the wing (the wing median line, named the chord line). This angle is named the angle of attack. The angle of attack obviously depends on the aircraft attitude (pitch angle).
When landing a tricycle aircraft the pitch angle must be relatively high, so that the nose gear doesn't touch the runway before the main gear. At the same time a fair amount of lift is generated due to the correspondingly relatively high angle of attack, as depicted in the left hand side of this picture:
In contrary, if we wanted to touch with the nose gear, we would need to strongly decrease the pitch angle. Such attitude automatically decreases the angle of attack, and the associated lift (right hand side above) which is already low due to the low speed required for landing.
If the lift is insufficient to counteract the gravity acceleration, the aircraft descends too quickly, and the vertical speed at touchdown creates a force on the gear exceeding its mechanical strength. Damages occur.
This is the reason airliners, which are tricycle aircraft, don't land on the nose wheel. However this leads to another question: Why do we use tricycles instead of taildraggers? Looking at the first picture, we see it would be uneasy to board and deplane, due to the floor slope, but there is more to know, maybe another good question to ask.