In general, relativity refers to the relationship between physical quantities measured in different frames of reference. Different relations are used depending on the desired accuracy. Relativistic effects in Star Control I and Star Control II appear in two forms: collisions, and projectiles.
The two common relativistic relations are Newtonian Mechanics and Einstein's theory of relativity. Newtonian Mechanics uses the Galilean Transform to add velocities. Thus a projectile's speed in the frame of reference b is simply the addition of the velocity of the source's frame a in b and the projectile's velocity in the source's frame a:
u' = u + v
where u' is the projectile speed in frame b, u is the projectile speed in a, and v is the relative speed between a and b.
This is accurate as long as none of the velocities don't get much above ten percent the speed of light. For speeds approaching luminal speed, the Special Theory of Relativity more accurately describes the addition of velocities by using a Lorentzian Transformation. It uses the following formula to add velocities:
u' = (u + v) / (1 + (u * v) / c2)
where c is the speed of light. For small u and v with respect to c, this reduces to the Galilean Transform. For collisions, both sets of physics agree that total momentum is conserved, but differ on the definition of momentum.
In the Star Control universe, collisions between ships and other objects appear to be not completely elastic and are governed by rules that more closely resemble Newtonian Mechanics. Projectiles however obey a different set of rules entirely. The velocity of a projectile is constant in the absolute frame of reference defined by the planet and the background starfield, no matter how fast the firing starship is firing. This can be verified with a Chenjesu Broodhome. The time it takes the photon crystal shard to wrap all the way around should be the same, regardless of whether the Broodhome is stationary, is moving in the same direction as the shard, or is moving in the opposite direction of the shard.
For some ships, this strange bit of physics has huge implications. A notable example of this is Mycon Podship, which when moving at top speed can overtake and be damaged by its own plasmoids. This is also responsible for the illusions of a longer range when firing backwards and a shorter range when firing forward.