Picard is beamed off the ship by an unknown force and put in a holding cell with three other prisoners exhibiting widely varying personalities. Meanwhile, a perfect doppelganger of Picard is put aboard the Enterprise in his place and impersonates the captain. The false Picard alters the Enterprise's course and takes it on an extended detour away from its scheduled rendezvous. The real Picard attempts to maintain control of an uneasy room where the tension between the prisoners continues to escalate as they try, and fail, to find a way to escape.
"Allegiance" is about observing behavior, and to be sure, there's some interesting behavior to observe here, particularly involving the faux Picard. He goes about his role as captain, and everything is just slightly off, but faux-Picard's explanations and reassurances to the crew are so pleasantly convincing that it's not obvious to the crew for a long time that there's something truly wrong here. The oddest of the behavior involves Picard's dinner with Crusher in his quarters (and she's at a loss to explain what happens there), and especially when Picard buys a round of drinks for everyone in Ten-Forward. The most entertaining moments in the show are watching the crew's confused reactions to the faux-Picard's slowly (but surely) increasingly odd behavior.
Of considerably less interest are the scenes in the mysterious holding cell, which document too little progress and quickly grow repetitive. We've got an annoying bully monster-alien in the room who threatens everybody (Reiner Schone), an annoying pacifist alien who acts like a smug wimp (Stephen Markel), and a Bolian Starfleet cadet (Joycelyn O'Brien) who remains mostly neutral. Picard must maintain a balance and keep the peace in this room. The clue that leads him to his discovery of the truth is pretty lame.
The whole episode has a palpable what-the-hell-is-going-on-here vibe to it, but it's a very slow burn that takes a long time (too long) to reach its climax, which is less than satisfying. The problem with the episode is that the conclusion (which is actually the story's underlying premise) in retrospect cannot support what happens in the episode. If the aliens responsible are simply trying to study the concept of leadership, how can they be so good at faking competent leadership with their copy of Picard? In fact, how can they be so good at copying Picard at all? I don't buy it.