How would an airplane land if the on-board radio breaks down?
It is important to remember that it is often difficult to diagnose radio malfunction during flight. Therefore it is hard to know whether you have a malfunctioning transmitter, a malfunctioning receiver, or perhaps both. You also have the point-of-failure that is the headset you are using. For this reason it is quite usual to carry a spare headset and the first piece of diagnostics is to swap headsets. Many General aviation aircraft also have a handheld microphone and speakers integrated into the radio stack.
Once you've ascertained that its the actual transmitter and/or receiver that has failed it is common to set the transponder to 7600. For landing, the relevant detail for the procedure to follow is contained within Chapter 4, Section 2-13 of the AIM:
Receiver Inoperative

Remain outside class D airspace until the direction of air traffic can be ascertained
Notify the tower your type, position, altitude, intention to land and request you are controlled with light signals
At 3 -5 miles, advise the tower of your position and join the traffic pattern
Watch the tower for light signals
Continue to advise of position (downwind, turning base, etc)

Transmitter inoperative

Remain outside class D airspace until the direction of air traffic can be ascertained
Monitor the primary local control frequency as depicted on sectional charts for landing or traffic info.
Look out for light signals that may be directed to your aircraft
During hours of daylight, acknowledge tower transmissions or light signals by rocking your wings
At night, acknowledge by blinking landing of navigation lights

Transmitter &amp; Receiver inoperative

Remain outside class D airspace until the direction of air traffic can be ascertained
Maintain visual contact with the tower and acknowledge light signals as above.

The light signals used for an aircraft in flight are as followssource:
Steady green - Cleared to land
Flashing green - Cleared to approach airport, or return to land
Steady red - Continue circling, give way to other aircraft
Flashing red - Airport unsafe, do not land
Alternating red and green - Exercise extreme caution
They can be seen in this video.