The original series ran for three seasons, detailing three of the five years of the U.S.S. Enterprise NCC-1701's five year mission. TOS is set in the late 23rd century, before any of the movies, roughly a hundred years after ENT and a hundred years before TNG.
In TOS, ships are fast, legs are visible, and time travel is intimately related to FTL drive. Starfleet is small - when it is "Starfleet" and not some other organization. The Enterprise is considered an "Earth" ship in truth; the fleet to which the Enterprise belongs seems to truly belong to Earth, although it acts with respect to the Federation as a whole. Weapons, when they are seen, are quite powerful; shields have curious vulnerabilities, but enormous strength against brute force, while STL travel is limited.
Space is a wide open place, full of more advanced beings as well as primitives, many of whom court self-annihilation. Computers often go wild, and are vulnerable to errant philosophy. The galaxy is acessible. Occasionally, the plot stretches beyond the galaxy, which is surrounded by an energy barrier for some unknown reason.