I've also found that you can generally have good success introducing non-gamer friends to games like Carcassonne, Settlers of Catan, or Ticket to Ride. They are games that don't have overly complex rules, are competitive but don't have to be cutthroat, and introduce them to a world where games are made for adults with more complex strategies to them. I've also had a lot of luck with cooperative games like Pandemic and Zombicide. Taking away the competitive nature of games can make it easier for folks that don't want to feel like they are at a disadvantage as a non-gamer. Plus the more experienced players can help ease them into their decisions without seeming self serving. You would be surprised at some of the people that I've been able to turn into board game geeks by just easing them into some of the classic euro-style games.
In the spirit of this thread, my gaming group recently started playing Codenames and found it to be a fantastic game. It's more of a party game, but was a blast. This gaming group is usually comprised of 3 couples, so we often play dudes vs ladies team games mixed in with our other games. Codenames worked great for this and everyone had a blast playing it. I'm pretty sure it's been around for a while, but I have to give credit to Wil Wheaton and the "Tabletop" crew for bringing it to my attention. On a related note, if you don't know about the "Tabletop" series, I highly recommend it. Wil Wheaton, of Star Trek TNG and Stand By Me fame, hosts 30-60 minute episodes where he and 3 or 4 of his actor/gaming friends play through a session of some great Tabletop games. I have to admit that I was introduced to probably half of my board game library through this medium. He explains the rules and then they play through the game. It's a fantastic way to 'play through' the game via proxy before committing to purchasing the game. I know I'm probably not alone in the trepidation at dropping $50-$100 on a board game at the FLGS without knowing if it will see more than one play through. This series allows you to see the gameplay and understand some of the rules (another priceless point since understanding the rule book can sometimes be a task in itself) before purchase. You can check out the episodes on the Geek and Sundry YouTube channel, or just search your favorite game with the keyword Tabletop and it will likely show up.