This is an overview of the various options for rules sets to use for Star Trek roleplaying.
This is a game from 1982, which has been out-of-print since the early 90's. Naturally, it is based on the original TV series, though there were a handful of sourcebooks for the movie era and Next Generation which were added as an afterthought. The background is well-developed, but it frequently added to the canonical background with its own explanations, which makes it more complete but often conflicts with later sources. Various sourcebooks added large tracts to the history and culture -- notably the Klingon, Romulan, and Orion sourcebooks.
Mechanically, the system uses percentile attributes and skills, where the basic roll is simple success/failure (i.e. not degree of success). Combat is rather complicated by an action point system and movement points (on a tile). It featured a "Luck" attribute as a balancing feature: Vulcans were better than Humans but less lucky, for example.
This is an original game published in 1993, also based on the original series, albeit indirectly. It is licensed by Paramount to use certain Original Series terms. However, the original rights came from before Paramount owned Star Trek, based on the "Star Fleet Technical Manual" of the original series which had an independent copyright. This really wants to be a 20th-century Special Ops marine game, and rather seriously distorts the background to get there. It focusses on elite "Prime Teams" which get beamed down to deal with specific problems.
Like FASA's effort, the background takes the original series and adds to it massively. It is unique in taking a fair amount from the animated series. For example, Niven wrote an animated episode which featured the Kzinti from his "Known Space" books -- thus these are in the PD universe as uneasy allies with the Federation.
Even moreso than the original Trek, the original PD emphasizes antiquated technology. The original series at least had phasers as a potent weapon which invariably downed the target with a single shot, barring only extreme circumstances like high-energy shields or silicon-based lifeforms. In PD they are only marginally better than modern bullets for damage. Also even moreso than the series, it tries to avoid the consequences of having a starship and transporters being on hand for the away team. Whereas the series would usually have a problem for the whole ship to deal with, PD tries to make the ship irrelevant so that the away team had to deal with their objective like an independent ops team.
It uses a dice-pool systems, taking the highest die out of a set of d6's where a 6 indicates an open-ended re-roll. Each task has a distinct "tri-code" indicating the three target numbers needed for minimum, marginal, and complete success.
There are a number of adaptations of GURPS for Star Trek role-playing, including an official adaptation of Prime Directive (see above) released in 2002.
There are also many unofficial GURPS adaptations, which tend to focus on TNG and later series and tend to not extrapolate much beyond canonical information. The following are links related to GURPS Star Trek, many taken from Brett Slocum's GURPS Conversions page.
This was a series of games which started in 1998 with the Next Generation RPG by authors Christian Moore, Ross Isaacs, Kenneth Hite, and Steve Long. It has a more rules-lite and dramatic approach than the FASA or TFG games. Actions are resolved using a "best roll" dice pool system, dubbed the "Icon" system. You roll a number of d6's equal to attribute (1-6), keeping the best die and adding skill (1-6). One of the dice in the pool is always the "drama die". If the drama die was a 6, you add it and the second-highest die. The total is compared against a difficulty number to determine success.
Character creation uses a mixed point-buy method. You get an initial template for your species, a package of basic skills for your department (i.e. Science, Engineering, Medical), and then a series of small skill packages for a series of steps in your characters background. As an option, you can simply spend from a large pool of points instead of the step-by-step packages.
This is in many ways a new edition of the Last Unicorn Games RPG, written by exactly the same authors for a different company. It is published in a Player's Guide and Narrator's Guide. It uses the "CODA" system, which is based on attribute + skill + 2d6 vs difficulty. Character creation uses templates and overlays for species and profession, along with limited point-based features such as advantages and disadvantages.