Within a Fleets, Squadrons are small groups of tight-knit Captains who work to further the prestige of the Squadron and it's Leader in Events.
A Squadron can be comprised of up to 5 Captains, one who whom assumes the role of Squadron Leader. In many respects, a Squadron functions similarly to a miniature Fleet, with the Leader being responsible for administrative duties. To administer or leave a Squadron, tap the Gear symbol to the left of the Squadron's name.
In addition, during certain Event types, the Leader can share one of their Crew members, allowing the other Squadron members to enlist that Crew for Event-related tasks. In return, the squadron Leader will receive a portion of the Victory Points earned through use of that Crew member. This can serve as a potent advantage for all involved.
Finally, at the end of an Event, your Squadron is evaluated for its overall performance just as you are individually as a Captain. Thus, membership in a Squadron that performs well overall will yield extra rewards.