What is the special aspect of Einstein's special theory of relativity?

The "special" part of Einstein's special theory of relativity is a reference to it being a special case of his General Theory of Relativity. Special relativity only applies in the special case where the curvature of spacetime is negligible, i.e., spacetime is "flat". Special relativity was formulated in 1907, and in order to correctly accommodate gravity, Einstein formulated general relativity in 1915.