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A Roemer delay is a delay in the appearance of a transient as observed from Earth, due to a greater distance between Earth and the phenomenon, in turn, due to the Earth's position in its orbit.
An example: when you check the observed times of the evenly-distributed pulses emitted by a pulsars, there is an observed delay (the Roemer delay) due to the Earth being at point of its orbit furthest from the pulser as compared to the times when the Earth is at the point nearer the pulsar. The EMR took longer to reach Earth. This delay-phenomena was first described centuries ago regarding observations of the orbits of Jupiter's moons over the course of years.
Analogous delays can be created at the source, e.g., EMR from an orbiting body, which can be relevant to pulsar timing of orbiting pulsars (e.g., analyzing PTA data). The term Roemer delay certainly was coined for such delays due to Earth's orbit (along with source orbits within the solar system) but possibly it is sometimes meant to include such effects of orbits of sources beyond the solar system.