Relativistic Doppler effect
The classical Doppler effect can be observed with any type of waves. When the source of the waves if moving towards the observer it causes the waves to bunch up, resulting in an apparently higher frequency. Similarly, if the source is receding from the observer, the waves are spread out and the frequency appears amaller. The effect of time dilation between a moving source and an observer complicates this situation (frequency is inverse time); the relativistic adjustment to the Doppler effect is called the relativistic Doppler effect.
Transverse Doppler effect
When the source is moving past the observer, displaced in the transverse direction the observed frequency is f = γf' at the moment the source is at closest approach to the observer and f = when the observer sees the source at closest approach.
Longitudinal Doppler effect
When the source is moving directly towards or away from the observer the frequency observed is:
f = f'
One twin remains on earth whilst the other makes a trip at high speed to a distant star and returns. Both twins, apparently, claim that the time of the other twin is dilated during the trip, and thus both claim that the other twin is younger upon reunion. This is not really a paradox. The twin who makes the trip has to undergo acceleration and deceleration which means that she is not in an inertial reference frame. Thus the stay-at-home twin is right and the traveling twin is younger than her sister when she returns.