Prime Directive is set in the Star Fleet Universe.
Prime Teams were a concept pioneered in the original Prime Directive game. The concept was a specialized ‘Landing Party’ that could do a wide variety of functions, including science, negotiations, and combat. The concept was developed by the designers who wished to avoid the television and movie cliché of having the Captain and primary bridge personnel always putting themselves in danger.
The following is a quote from the Prime Directive Source Book, 1st Edition, page 12:
Every starship has an officer in charge of landing parties and other detachments of personnel sent from the ship for various missions. This officer, known as the Briefing Officer, has several personnel assigned to his department and draws additional specialist from the ship’s other departments as required for a specific mission. In this regard, most landing parties (known as Action Teams) are temporary ‘task forces” organized for a specific mission and disbanded when it’s over.
Typical Action Team
Ships which consistently face this type of mission (e.g. most cruisers) and which have suitable personnel available often form more or less permanent Action teams which remain together for extended periods and have personnel with a sufficiently wide variety of skills to handle any (or at least most) missions.
When these teams reach a certain level of competence and cohesion, they may be recognized by Star Fleet as a Prime Team and will spend time at Prime central before returning to their Duty Assignment. They may then adopt a unit name, logo, etc. In some cases, Star Fleet (or the numbered tactical fleets) specifically form and train potential Prime Teams. Prime Teams are considered the elite of the Star Fleet special operations corps.
Typical Prime Team
Notes on differences between the SFB Universe and Star Trek: TOS
Obviously, Star Trek: TOS has some problems. There is no real effort to really look at the background until season two, for starters. They sometimes seemed to ignore some weapons and were not always consistent. SFB is a wargame, and as such, everything is spelled out and consistent. Prime Directive also must do this, to be a stable game.
This setting is using the Star Fleet Universe timeline, which does not perfectly match up with Star Trek. One glaring area of difference is the inclusion of more non-humans in Star Fleet. In TOS we only saw one non-human in Star Fleet, and he was a curiosity. (The cartoon series did give us the guy with the long neck and three arms, at the helm). In Prime Directive, while fully 70% of Star Fleet is composed of Humans, there are many other races in Star Fleet, especially those with Full Member status.
There are also events in the series, such as Balance of Terror where the Enterprise fights the Romulan ship. In the Star Fleet Universe, this is part of the larger Second Federation-Romulan war, and not resolved in a single episode.
The Future history can be found here.