Being a little on the older side (mid 40's) and a former active gamer, I tend to like the collectible or trading card games such as Magic the Gathering and Yu-Gi-Oh (I got into these when adult life got in the way of RPG's being an appropriate hobby) as well as the classic board and card games such as Chess, Checkers, Pente, Hearts, Backgammon, and Go.
As I tend to have some social anxiety with tendencies to isolate when I am not at work, I have greatly enjoyed the ability to play these on the computer. Since I've moved up to Xubuntu 12.10 on my machine at home they either have to be compatible with Linux, web-based, or Java-based (as these can be run with no problem) or very Wine-friendly.
For MTG (solitary vs AI when I am not playing real-world on the table against my son), my favorites include Magarena and Forge. Both are programmed in Java. For both you have to download card images separately, but that is not a problem as this is done within the program itself (just do it the first time you run them). There have been some great improvements and a vast expansion of available cards in the new versions of Magarena and Forge, and I found another fun Java-based trading card game called Dreamwalker. In terms of online trading card games, I'm playing Elements, Monster Master, and Rolplay.
I do admit also liking the Yu-Gi-Oh TCG and even the Pokemon TCG which I play on the computer using a Gameboy Emulator and ROM's on Ubuntu. I think I would feel, um, funny going to the game store to play against the normal crowd for these games, but they can be fun, short diversions. I like the combination of strategy and luck you can get with CCG's. Any other suggestions for browser (Flash) based, Java based, or native Linux (or Wine friendly) trading card games or traditional board or card games, etc.?
The one benefit of Ubuntu Linux is that there are good choices on classic games which can be run natively - obviously chess, checkers, backgammon, and hearts - but also some less familiar games like Go (although I do need to increase my base knowledge of this game), Mancala, Reversi, Pente, and a great game called Pasang Emas which is a really cool classic abstract strategy game from the island country of Brumei.
I also like Battle for Wesnoth as well as some simple "simulation" games like Dopewars, Project Football, and Bygfoot.
Other suggestions of similar games would be appreciated!