The term relativity is used for two theories developed by Albert Einstein that accommodate the apparent constancy of the speed of light, even to someone traveling fast in the same direction as the light beam. The constancy was evident in the Michelson-Morley experiment, which strove to find the velocity of Earth as compared to ether, the name given to whatever substance has the waves that we perceive as electromagnetic radiation. The inability to measure such a velocity by such means was a chink in the Newtonian paradigm, i.e., the laws of gravity and motion summarized by Isaac Newton.
The Lorentz transformation (aka Lorentz transform) was a mathematical attempt to explain the apparent constancy through effects of motion on the dimensions of objects. Einstein showed the transform was consistent with a model of nature that matched experiment, but at the cost of throwing away our preconceived notion of simultaneity, events occurring at the same instant in time: that whether two events are simultaneous depends on the relative motion of observers. The theory demands the Lorentz transform rather than the Galilean transformation (aka Galilean transform), which matches our intuitive senses and everyday experiences, but at typical everyday speeds (aka non-relativistic speeds), the two converge. (The adjective relativistic is often used specifically to specify a regime in which relativity plays a significant role, particularly, involving speeds near c, the speed of light in a vacuum).
This was termed relativity, then later special relativity when Einstein developed general relativity, which extends it, also modeling gravity.
Relativity shows how even though you and your surroundings may be in motion, that motion isn't evident unless you can observe something moving relative to you. In addition to Einstein's conceptions, the term relativity is also used for earlier explications of this concept, that didn't take into account factors evident more recently when Einstein tackled the problem, e.g., Galilean relativity.
While Einstein's relativity gives up the concepts of absolute time, simultaneity, and distance, identical in all frames of reference (i.e., the same no matter what your speed and direction while you measure), it does yield a minimal time or distance between events: