In the first few minutes of the episode, we see Picard, Riker, and Worf standing on the bridge with their arms behind themselves. Then they cut to Picard and Riker's backs and we see their arms at their sides.
When Picard goes to do the "Picard Maneuver" in the Stargazer, Data says that "the ship will seem to disappear," but what really happens is the ship will seem to appear in two places at once.
Usually only Picard calls Riker "Number One," but in this episode Dr. Crusher also calls him that.
During introductions between the Ferengi and Enterprise crew, Picard introduces Riker and Data. In between, the overhead mike bobs down into view.
When Picard beams over to the Stargazer the final time under mind control, DaiMon Bok raises the Stargazer's shields so as to prevent Captain Picard from being beamed out. Later in the same scene, however, DaiMon Bok then beams out so Picard can destroy himself.
The approach of a starship on long-range sensors shouldn't initiate, and never has initiated, an intruder alert.
Troi states that she senses great deception from the DaiMon of the Ferengi ship, though it was previously established (in "The Last Outpost") and later thoroughly established (everywhere) that Ferengi minds are closed to telepaths (and thus also to empaths).
Picard (incorrectly) introduces Data as his second-in-command. Data is the Enterprise's Second Officer, but Riker is the second-in-command.
Riker locks a tractor beam on the Stargazer with its shields up, even though it is established TNG-era tractor beams can't do that (see "The Best of Both Worlds" for an example).
Kazako: As you humans say: I'm all ears!
Picard: What is wrong with me?
Beverly: I wish the hell I knew, Captain. But something unusual has definitely been happening to you.
Picard: Why do doctors always say the obvious as if it's a revelation?
Beverly: Why do captains always act like they are immortal?
Picard: Let the dead rest, and the past remain the past.
Kazada: The android was mentioned. What is its price? We should like to purchase it.
Picard: He is not for sale. Commander Data is, um... is um...
Riker: ... is second-hand merchandise. And you wouldn't want him.
Data: Second-hand, sir? Oh, of course. A human joke.
Bok: That vessel is yours, if you wish to have it.
Rata: At no cost? Ugly, very ugly.
This is the first mention of the "Picard Manuever," which in this case refers to the battle tactic Picard used when the Stargazer was attacked by an unknown vessel; he jumped to high warp, making it appear to the attacking ship that the Stargazer was in two places at once. He was then able to destroy the unknown vessel. The "Picard Manuever" has since come to be known as the term for Patrick Stewart's habit of tugging at his costume when he sat down or stood up.
The USS Stargazer was initially supposed to be a Constitution-class starship, allowing the crew to use the movie version of the original Enterprise in the series. This is why a model of a Constitution-class starship can be seen in Picard's ready room in early episodes. However, it was later decided to create a new model for the Stargazer, but by the time they did, a scene on the Stargazer's bridge had already been filmed with Geordi referring to the ship as "Constitution-class." Hence, the term "Constellation-class" was developed, which was similar enough to "Constitution-class" that they could re-dub the scene without reshooting. Also, Picard's Constitution-class model was replaced in later episodes with a model of the Constellation-class Stargazer.
The Stargazer bridge is a redress of the Enterprise-D Battle Bridge set, which was originally the Enterprise movie bridge from Star Trek I-IV. The Stargazer turbolift interior was built for the first Star Trek movie. This is its last appearance in the Star Trek universe to date. Picard's Stargazer quarters are a redress of the Enterprise Junior Officer's Quarters, originally built as Kirk and Spock's quarters for Star Trek I and Star Trek II.