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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 that of ether, the name given to whatever substance has the waves that we perceive as electromagnetic radiation (EMR). The inability to measure an Earth velocity by such means was an anomaly 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 observed events are simultaneous depends on the relative motion of the observer to the two events. The theory demands the Lorentz transform rather than the Galilean transformation (aka Galilean transform). The latter matches our intuitive senses and everyday experiences and at typical everyday speeds (aka non-relativistic speeds), the two transforms 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). Einstein's theory was termed relativity, then later was called special relativity (SR) after he developed general relativity (GR), 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. The term relativity is now also commonly used for earlier explications of this concept, such as the term Galilean relativity, which didn't take into account factors that were evident when Einstein turned his attention the problem.
While Einstein's relativity gives up the concepts of absolute time, simultaneity, and distance that are 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:
Both special relativity and general relativity are deterministic: they calculate a deterministic future-prediction given a current state, just as do Newton's laws. Though this is the case, there have been people who associated Einstein's relativity with the idea that objective reality is somehow different for different people, a notion termed relativism. Relativity theories are unrelated to relativism and purport a single reality that everyone lives within.