TOS takes place 200 years into the future, where Earth is united and prosperous, and heads up a "Federation" of worlds which freely share technology and culture. They also try to protect planets whose inhabitants haven't made it into space yet. To reach the stars, one needs "warp drive," which allows faster-than-light travel. (For "hyperspace," you need Star Wars.) The Federation's "Prime Directive" forbids people from going to a "pre-warp" planet and altering their natural development.

The Federation's space military is called "Starfleet." It runs its ships like our modern navies do, with large cruiser-sized ships (e.g. the Enterprise) which carry smaller ships, which range from little "pods," to medium-sized "shuttles," to larger "run-abouts." Starfleet has some all-out military ships, but also has medical and scientific ships.

The primary races in the Star Trek universe (other than Human) include:

Vulcans - Allies with Humans, they are as recognizable by their pointy ears as they are with their mastery of logic and absence of emotion

Klingons - The primary enemy in TOS.  A fierce warrior civilizaton that lives by a strict honour code

Romulans - Distant relative to the Vulcans, also an enemy to Starfleet

Ferengi - A race driven by commerce, they are always looking to maximize their profits (would probably make excellent business students!)