The Doppler compensator is the part of the transporter system in Star Trek that compensates for relative motion of the origin and the destination of the transport. Without this device, the Doppler effect would lead to a wavelength shift - the Doppler Shift - which would distort the transport pattern. The matter stream is briefly stored in a pattern buffer while the system compensates for Doppler shift to the destination, giving the Doppler compensators time to adjust the transporter's targeting scanners.  The matter stream is then transmitted to its destination. In the Star Trek: Deep Space Nine episode "What You Leave Behind", Captain Benjamin Sisko could actually hear the USS Defiant's doppler compensators being out of phase.

The Doppler Shift is a real physics phenomena that you can observe yourself - note how a vehicle siren's pitch changes as the vehicle raced towards, then away from you. First the pitch became higher, then lower. This change in pitch results from a shift in the frequency of the sound waves - the Doppler Shift. As the vehicle approaches, the sound waves from its siren are compressed towards the observer. The intervals between waves diminish, which translates into an increase in frequency or pitch. As the vehicle recedes, the sound waves are stretched relative to the observer, causing the siren's pitch to decrease. The electromagnetic radiation emitted by a moving object also exhibits the Doppler effect. The radiation emitted by an object moving toward an observer is squeezed; its frequency appears to increase and is therefore said to be blueshifted. In contrast, the radiation emitted by an object moving away is stretched or redshifted. Blueshifts and redshifts exhibited by stars, galaxies and gas clouds  are used in astronomy to calculate precisely how fast they are moving toward or away from Earth.

The effect of Doppler Shift on spectral lines is used
to calculate how fast astronomical objects are
moving away or towards the observer