An inertial reference frame is a coordinate system in which the laws of physics take their simplest form. Specifically, it is a frame in which a body subject to no net force moves in a straight line at constant velocity — or remains at rest.
The surface of the Earth is approximately inertial for most purposes, though not exactly — the Earth rotates, which introduces small fictitious forces (Coriolis, centrifugal). For the purposes of this theory, we deal with ideal inertial frames unless otherwise stated.
Two inertial frames related by a constant velocity are both inertial. This is the content of the Principle of Relativity — no experiment conducted within either frame can determine which is “truly” at rest.
In standard notation, the home frame is designated S and a frame moving relative to it at velocity v is designated S′.