Details on the crew and bridge officers in Star Trek Online have been revealed in an update published on the Star Trek Online website.
Crew is basically a ship statistic representing your ships "red shirts" and the amount of crew your ship has determines the efficiency with which a variety of tasks are performed. Crew will be required for repairing hull or subsystem damage, repelling boarding parties, attacking on boarding operations (which have now been revealed to serve the function of damaging an enemy ships subsystems), and more. The amount of crew you have on hand will determine how effective they are at performing these tasks, and they can be injured or killed in battle. Crew members that are injured will become unavailable until they are fully healed in your ships sick bay, and the rate at which they are healed and put back into service is dependent on your ships medical equipment and your Science Officers' skills.
Each ship class can hold a different amount of crew, with the smallest ships being able to hold as little as 50 crew, while large cruisers can hold crew of up to 1,000.
Details on the Bridge Officers role in gameplay are a bit less specific, though we are assured they will play a "key role" in ship operations. Each ship will have a variety of stations of different types available to it, with different ship classes favoring differing focuses on their allotment of stations. For example, a Science vessel may be comprised of mostly Science stations, while Escort ships focus on Tactical stations, and Cruisers on Engineering stations.
Naturally, a Science Officer can only man a Science station and so forth. There is another caveat, however. Each station will also have a maximum rank associated with it, and a Bridge Officer accessing a station that is lower ranked than he is will only be able to utilize his skills earned up to the rank of the station. For example, a Science officer at Commander rank would only be able to access his skills up to the Lieutenant rank if he accesses a Lieutenant ranked station.