I use board games as a camp activity rather than in classes, but they've been moderately successful.  I've used Splendor and Carcassone and the kids picked them up fairly well.  Catan was a bit too difficult for the kids I tried it on, at least in English, but more advanced speakers could do it.  For very low-level speakers, I use UNO.

English practice is usually provided by making the students announce all of their game actions.  So UNO practices "I have a red 5."  Splendor uses phrases like "I buy this card with 5 red gems and 3 black gems."  Carcassone has things like "I will add to this road," or "I've finished this city."

It would be tough to tailor games like that to particular lessons, I think, but as a general conversation practice camp activity, they work well.

Uno is great of course.

Also we just played this as an intro ice breaker lesson but you can surely integrate lesson review into it.

The kids had so much fun playing it. My family always play it at christmas too.

https://en.wikipedia.org/wiki/Tippit

What I like about this game is that you can play with them and they can outsmart you.

As far as games that are good for English, Bohnanza is a hit with my kids. They absolutely love it. FUSE also works if you have a small group. Abracawhat is good for practicing small numbers. Higher level kids enjoy One Night Ultimate Werewolf.

I also have a bunch of other games that my kids can play on their free time. I have a lunchtime crew that hangs out in the English class and plays Bang! Other kids like Bang Dice (which I prefer but that group of kids doesn't).

I actually have quite a budget dedicated to exclusively buying board games. I think I'm kinda lucky.

I typically have two kinds of kids: the really smart kids who want something a bit more in-depth, and the kids who just wanna play something brainless. I try to cater to both groups. What I typically do is teach all the game rules and English expressions to my upper level kids in my after school class, and they end up teaching the lower level kids in the board game club the rule details I can't explain.

To the low English learners: Twister, Monkey Drop (it's like Ker-Plunk, but with monkeys), Ultimate Werewolf, Sorry!, Uno, Battleship, and they go nuts for Halli-Galli. For the upper level kids: Catan, Bohnanza, Timeline, Bang!, Dixit, Rummikub, the DaVinci Code, Splendor, and they particularly love the Bang Dice Game. The big dud of the games that I bought was King of Tokyo. It's pretty much shunned.

You should be aware of one website, which I've been able to find Korean instructions for games that are only in English: http://www.bgkorea.com/  It appears to sorta be the boardgamegeek of Korea. Also, don't be afraid to email developers of these games. The King of Tokyo guys were great... I had a Korean edition of the game and didn't understand the cards. One email later, and they sent me a PDF with the English scans ready to go.

Lookin forward to what other people play. I need more ideas for new games this year :)

We played lord of the rings themed monopoly the other week, it was really fun if the kids are looking for something more complicated.
We taught can, must, may and have to.

I have used many table top games, but THE HANDS DOWN BEST ONE IS Pepper's Talking Board

The game is not cheap, so be sure to use the school's budget. I had a camp budget, and I bought a few of the board games, and made everything else for camp without using any of the budget. This game makes students speak...A LOT. They have a few clips on Youtube, but the lady speaks a lot of Korean in the videos, I guess for the sake of making the video more fluid(?).  Anyway, it is great for speaking classes. If you do check it out, reply back with how awesome it is! (or what you thought of it  )

Oh yeah, high level 3rd grade middle schoolers also love Catch Phrase (the actual game. It can be as high as 80,000 with shipping). I got two, and we split off into two groups with about 10-14 per group. It works well, and they have a blast once they get the hang of it. Since the timer is a bit fast, I sometimes tell them to do two rounds on the timer.