Relativity is the study of the geometry of events in spacetime. An event is a position in three-dimensional space, along with a time. Euclidean geometry describes that space, but time is very different, as you can go back to the same position but not the same time. Relativity puts a non-euclidean geometry on spacetime that limits the speed of causality. Events can only affect nearby events if a light signal or slower signal can travel from one event to the other.
The first relativistic theory was devised by Maxwell in 1865. His electromagnetism theory used fields that propagate at the speed of light. This was in contrast to gravity, where similar inverse-square laws seem to act at a distance. He wondered whether this theory could detect the motion of the Earth. The Michelson-Morley experiment failed to do that in 1887, and seemed to show that the speed of light and the rest of Maxwell's theory was the same under inertial motion. Lorentz explained this by showing that motion modifies how we measure space and time.
Poincare and Minkowski showed in 1905-7 that there is a non-euclidean geometry on spacetime that underlies Maxwell's theory and explains the experiments. The theory can be expressed in equations that are covariant under the geometric symmetries. That geometric view came to dominate XX century Physics. It was important because it changed our understanding of space and time.
Einstein's chief contribution was to show that the Lorentz transformations could be derived from the principles that Lorentz and Poincare deduced from Maxwell and experiments. He never really accepted the geometric view.
Special relativity is the infinitesimal version of general relativity. Like a tangent line is the infinitesimal version of a smooth curve. In special relativity, the spatial part of spacetime is Euclidean, ie flat. Spacetime is curved in general relativity, and the Ricci curvature tensor is essentially the mass-energy.
When people say that relativity is non-euclidean, they usually mean that spacetime is curved from the gravity of mass. But the flat no-gravity tangents are also non-euclidean, because of the causality speed limit.
In my opinion, this non-euclidean geometry of flat special relativity spacetime is the true essence of relativity. It was what Minkowski explained in his wildly popular 1908 paper. That paper, along with experimental confirmations, is what sold physicists on relativity theory. Poincare's 1905 papers also had it, but were probably only understood by Minkowski and a few other mathematicians.
Most relativity historians ignore this essence, but see The Non-Euclidean Style of Minkowskian Relativity, by Scott Walter. It explains how Planck, Wien, Laub, Sommerfeld, Einstein, and German Physics journals all rejected the non-euclidean geometry approach around 1910-12. By then, Poincare and Minkowski were dead, and the biggest champion of it was Varicak.
General relativity was the natural generalization of special relativity, once the tools of Riemannian geometry were developed. Einstein's main contribution was to calculate the deflection of starlight during a solar eclipse, and the precession of Mercury's orbit.