Different types of waves can be used for communication in many ways. The simplest way could be as communication is done using the Morse code, where only the presence or absence of a wave is used to interpret the signal that is being sent.
A more advanced way of communication using waves is by a process called modulation. Here one wave is used as a carrier and its properties are varied according to the signal that has to be transmitted. Modulation can be done in terms of a change in frequency, change in amplitude, change in the phase, etc.
The waves that can be used for communication can be electromagnetic waves, sound waves, etc.
This is a pretty broad question as there are many types of waves and many forms of communication. In essence, communication is the transfer of some kind of information. All waves can be described by a simple formula:
f(x) = A cos( wx + p ), where A is the amplitude of the wave, w is the frequency of the wave, and p is the phase of the wave. Sound waves, deep ocean waves, and electromagnetic waves can all be represented by this equation. To transmit information on the wave, it has to be modulated.
Modulation means that you are changing some component of the wave in order to convey some information. For example, when we speak, our vocal cords vibrate to create a sound wave. The frequency of vibration is less important in forming words than are the changes we make to the sound with our lips and tongue.
An electromagnetic wave can be used to communicate by changing the amplitude of the wave from high to low to transmit binary ones and zeros. This is called Amplitude Modulation (AM).
Or an electromagnetic wave can be used to communicate by alternating between a faster and slower frequency. This is called Frequency Modulation (FM).
Finally, an electromagnetic wave can have information encoded into the phase of the signal. This is called Phase Modulation (PM). It is not as widely used as FM and AM because it is difficult to implement from a hardware standpoint, and suffers from poor efficiency.