Below you'll find lists of what I consider to be some of the best iOS, Android, and PC board-game experiences. But all of these experiences are much better with Friends, so if you're on iOS feel free to look at the NeoGAF board-game community spreadsheet to find people to add. If you're looking to try things out on other platforms, feel free to use the thread to see if you can find folks to join you. Synchronous play can be a bit tough to coordinate but thankfully about 85% of these games provide adult-friendly asynchronous modes.
We also often have GAF tournaments in some of the most popular games, so keep an eye on the thread for those. Without further ado, the lists!
Best iOS Board Games
1. Carcassonne: The original and still the best. A simple tile-laying game with hidden depth (especially with expansions) and excellent asynchronous multiplayer.
2. Agricola: More in the deep end in terms of complexity, but an incredible adaptation of one of the best euro games of all time. Manage your family and work assignments and try to build up your farm through improvements and occupations while avoiding ever-looming starvation. Top-notch async as with all Playdek titles.
3. Lords of Waterdeep: A lighter worker-placement game with a D&D theme. Great intro to euro-style games for people coming from D&D, Magic, or more fantasy-themed gaming. Done by Playdek so async is great.
4. Summoner Wars: Two player mediumweight fantasy strategy war game. Construct a deck of minions and put them into battle on the board. Really quite for a quick wargaming-type fix without all the rules complexity, but with cool fantasy races and abilities.
5. Lost Cities: Made by the Carcassonne gang, an even lighter two-player card game that involves taking gambles on certain suits as you try to play cards incrementally to each suit. Like Carcassonne, deceptively deep and fun.
Honorable Mentions: Eclipse, Small World 2, Neuroshima Hex, Yomi
Best Android Board Games
x. Summoner Wars: Same as iOS version, and there's even cross-platform! Probably the top game on Android now.
1. Small World 2: Risk uber-lite and quick in a goofy fantasy setting.
2. Neuroshima Hex: Two player hex-based wargame with variable player and unit powers. Similar to Summoner Wars but with about an extra half-step of complexity and a nice cyberpunk theme.
3. Ticket to Ride: The classic simple gateway game in app form! Collect suits of train cards and play them to the board to create routes and complete secret missions.
4. Carcassonne: Not quite as good as the iOS version (totally different developer), but increasingly coming into its own. Definitely worth it if you don't already have it on iOS.
5. Alhambra: A little bit on the kludgy side in terms of implementation, but the game is a great one where you collect currency to buy buildings and build up your city.
Honorable Mentions: Mr. Jack Pocket, Kingdom Builder
Best PC Board Games
While the below are among the best individual applications for PC, PC actually has the best selection of digital board games on websites that host many games: boardgamearena (Seasons, Race for the Galaxy, Puerto Rico), yucata (Stone Age, Roll through the Ages, Balloon Cup), and boiteajeux (Agricola, Dungeon Petz, Tzolk'in)
1. Memoir 44 Online: Great implementation of the classic super-light wargame that introduced many eurogamers to wargaming.
2. Ticket to Ride: See above.
3. Small World 2: See above.