While it's generally best to watch a show in chronological order, so you can see the characters develop, there are usually episodes in the run of a series that are self-contained, and little to no background is required to enjoy them. These are the ones I'd like to focus on. In each case, there are others that might do just as good a job, but these are my picks.
Star Trek: The Next Generation
Star Trek: The Next Generation has the advantage of having many self-contained episodes and few multi-episode spanning story arcs. This makes the show easy to pick up and enjoy. But when I think of the series, Darmok is usually the first episode I think of.
The execution of the theme is original, and totally what Star Trek is all about. This story is fully diplomatic, it's just a matter of learning to communicate. On one side is hand-to-hand combat with a shiny beast, and also stories around the campfire. On the other side is... well, research. Half this episode is just characters researching language, stories, and discussing their findings.
All you need know is the barest information about what the Federation is, and how they operate. And not necessarily even that.
Buffy the Vampire Slayer
Information about the characters might help this episode, but it's true value is in the direction and cinematography, which can be appreciated on their own merits.
This is one of those episodes. It's self contained, well designed, really a fantastic episode. Frankly, it's such a good episode that it might even spoil the series a little. This is definitely the first episode I show anyone who is curious about Dr. Who. Even if they don't end up liking the series, they will more than likely enjoy this episode. It is well-executed, original and completely self contained. The lead roles are one-off characters. Outsiders, like the first time viewers, unfamiliar with the series they inhabit.