The Enterprise provides transport for the ambassador (Tim O'Connor) of a world that's attempting to negotiate with a neighboring system the end of their long-standing conflict by way of special negotiations and reconciliation ceremonies. The ambassador has brought aboard mysterious cargo that will be used in these negotiations. It turns out the cargo is actually an empathic metamorph — an exotic woman named Kamala (Famke Janssen) who has the unique ability to sense what a mate wants and be exactly what he wants her to be.
Her entire life has been in preparation for an arranged marriage; she is to be a gift in these negotiations (though the episode is very clear that she does this of her own free will and is not a slave). Kamala was supposed to spend the trip in stasis, but the meddling of some mischievous Ferengi releases her prematurely, leaving her with several days to spend aboard the Enterprise, where she attracts a great deal of attention.
First, let's get this out of the way: The Ferengi (including one played by Max Grodenchik, the future Rom) are an egregiously dumb piece of business here. They provide their typically annoying and contrived role. Why are they so easily able to access the cargo bay, which is supposed to be under lock and key because of the special cargo? Simply because the plot requires it, just like it requires the Ferengi later to bumble their way into accidentally knocking down the ambassador and putting him in a coma, requiring Picard to step in and carry out the negotiations.
But forget about the Ferengi. "The Perfect Mate" is really a story about Picard, and the relationship that he has — or, more to the point, doesn't have — with Kamala in the course of the days leading up to the arranged marriage. Kamala's diverse education has prepared her for all possibilities, and her empathic nature allows her to adapt to almost any situation. What's interesting here is not just that Picard finds that Kamala may actually be exactly the kind of woman he needs, but that Picard may be exactly what Kamala needs in return. The fact that Kamala will miss out on a much richer, fuller life aboard the Enterprise, instead being a diplomat's trophy wife, is not lost on her.
What we have here are two people trapped between their duty to the greater good and the possibility of a meaningful relationship. And what most struck me about this episode was the unspoken but clearly conveyed message that a man like Picard is essentially destined to a life of solitude because his responsibilities as a starship captain do not allow him to make a relationship a priority. Here's a man who's the paragon of integrity, sometimes at the expense of his own personal happiness. That all sounds like a cliché, but in its quietly effective and unassuming way, this is a character study examining that sacrifice.