So Wil Wheaton has a YouTube show about boardgames, and it's awesome. It's called TableTop, and it's a long-form (about a half hour per episode) look at various good boardgames, including a basic explanation of how to play each -- plus an actual play session so you can actually get it. Whether you're a boardgame nerd or a "normal" person looking for a fun diversion for your next party, this is for you. I've collected five starter episodes for your enjoyment. The TableTop YouTube channel has tons of outtakes and bonus features for each episode.
Settlers of Catan
Wil explains how Settlers is an "infection vector" to introduce non-gamers to tabletop gaming. Jane Espenson (!), James Kyson, and Neil Grayston play through and learn. I also learned that I've been pronouncing "Catan" incorrectly for years.
Ticket to Ride
Ticket to Ride is reasonably easy to play, and you might just learn some geography (especially if you buy the European version). In this episode, Wil is joined by his wife Anne, as well as Colin Ferguson and Amy Dallen.
Now here's one I haven't played! Munchkin is Steve Jackon's parody of role-playing games involving cards and cooperation. Felicia Day (!), Steve Jackson (!!!), and Sandeep Parikh join Wil.
Small World is kind of a geekier version of Risk. By adding powers to each player's race, the gameplay changes a lot between games. Grant Imahara (!), Jenna Busch, and Sean Plott (aka Day) join Wil to dominate the table-sized world.
(Thanks to reader Kim Krzebiot for introducing this show to me!)