Sound waves are often characterized by four basic qualities, though many more are related:
Frequency, Amplitude, Wave shape and Phase*
Some sound waves are periodic, in that the change from equilibrium (average atmospheric pressure) to maximum compression to maximum rarefaction back to equilibrium is repetitive. The 'round trip' back to the starting point just described is called a cycle. Periodic motion is classically demonstrated by the up and down motion of a dropped weight (mass) attached to a spring or by observing the motion of a pendulum. The amount of time a single cycle takes is called a period.
It is possible to measure frequency in seconds per cycle or periods, but it is far more common for sound measurements to use cycles per second.
Periodic motion depends on two prime factors; 1) elasticity, in that medium being distorted return to its original state (equilibrium), and 2) a source of energy to initiate and sustain motion. In the case of sound waves, the atmospheric pressure will return to the ambient pressure without an energy source to disturb it, and any vibrating surface will constitute an energy or excitation source.
Simple harmonic motion, the motion described by mass/spring example above, is represented in sound as a sine wave, which traces the mathematical shape of it namesake. A sinusoidal wave (which also includes a cosine wave) is the only wave shape that produces a singles frequency, as we will see in the waveform chapter. With any minute deviations in the sine shape, additional frequencies will be generated.
Noise is characterized as being aperiodic or having a non-repetitive pattern. There are many different types of noise, depending primarily on the random distribution of frequencies. For example, some types of noise may sound brighter than others.
Some periodic waveforms can be complex enough to be perceived as noise if our ears cannot detect perceptible pitches. Many real-world sounds, such as the "chiffy" attack of a flute note contain some combination of periodic and aperiodic components.
*It could be argued that phase is not a characteristic of a single wave, but only as a comparison between two or more waves.
For further study, see Hyperphysics->Periodic Motion