Back in the 1960s, heck even the '70s or '80s, the name "Harry Mudd" would have been one of the most famous ones in Star Trek lore. Today, not so much.
Mudd made a handful of appearances in the series and even The Animated Series. He is without question a classic figure in franchise lore, but his importance rapidly diminished over the years. He certainly didn't appear in the films. He did make it into Star Trek Into Darkness, but only as an oblique reference (the prequel comic book adds a good bit more, but then, that only counts so much). His presence had no direct effect in any of the later series, unless you count someone like Okona or Kasidy Yates. There! Count both of them!
Basically, Mudd was a lovable scoundrel. He was also a jolly fat man. If he'd appeared in "The Trouble with Tribbles" handing the little fuzzball to Uhura, his legacy would be assured. Instead he dealt in androids.
In androids??? You'd be surprised how often androids showed up in the original series. That alone is a whole subgenre that helped usher in the era of Mr. Data. Mudd's corner of this subgenre originally began as a con. "Mudd's Women" doesn't feature androids, but his efforts to create perfect female companionship also prefigures Quark's holosuite programs (remember the one with Kira?) in Deep Space Nine.
Mudd is one of the first civilians to meddle in Kirk's affairs. Usually in Star Trek the people who aren't in Starfleet or belong to adversarial alien species are the residents from the planet-of-the-week. In that sense, Mudd is also a predecessor to the space station in DS9, filled with a whole society of its own.
As such, I recommend this episode as essential to the series if not directly the whole franchise. Mudd sets a precedent, even if you don't remember or particularly care for the character yourself. You can't watch or enjoy the original series without including Mudd in its legacy. And now you know why.
franchise * series * essential * character
Roger C. Carmel
Memory Alpha summary