Picard meets the ship's new head of stellar science, Lt. Commander Nella Daren (Wendy Hughes), a smart, strong-willed woman and talented piano player, and he slowly learns that he enjoys spending time with her. They have deep conversations. They enjoy playing music together. Picard realizes he might want to have a real relationship with this woman — a member of the crew — which is not something he takes lightly. He tiptoes around it for a while before realizing that it might be something he wants to seriously pursue.
"Lessons" succeeds where so many TNG would-be romances have failed because it considers the romance as a serious and realistic piece of business and not as a hopelessly arbitrary and unconvincing afterthought of the plot ("Aquiel," "Birthright, Part II"). Here is the Starfleet equivalent of an office romance; Picard and Daren must proceed cautiously, because he's the captain, she is a member of his crew, appearances matter, and there are plenty of people who could potentially be made uncomfortable with the situation, even if no one does anything wrong. (Riker has such a moment where he questions whether his objectivity is being affected with regard to Daren in light of her relationship with Picard.)
Also important is how the story spends the necessary time setting up the relationship to give it legitimacy. Daren and Picard share an interest in music, which leads to a number of nice scenes featuring classical music, including one in the ship's most acoustically perfect location. The music lessons take on even more meaning when Picard explains to Daren the story behind the flute he plays, which serves as a welcome callback to "The Inner Light" and lends a lot of credence to the story's emotional center.
Ultimately, this story's lesson covers familiar territory similarly mined in "The Perfect Mate" — Picard cannot avoid a life of solitude because he will always have to choose duty over companionship. This theme reveals itself in the closing acts, where a crisis arises and Daren must be sent on a dangerous mission where she nearly perishes, forcing them both to confront the reality they both probably knew was already there. Naturally, Picard is not about to stop being the captain of the Enterprise, and TNG is not about to take on a permanent girlfriend for him. But "Lessons" presents a one-off romance with solid execution, believable situations, good performances from Wendy Hughes and (naturally) Patrick Stewart, and a genuine emotional core.