By faaaaar my favourite 2 player card game is called Spite and Malice (also apparently called Cat and Mouse sometimes) and has been a family staple for years in my household.
You can easily find out how to play by punching the name into the usual search engines or Wiki so I'll just assure you that the game is in-depth (games usually last about 40-80mins depending on hands you are dealt), involves two decks and lot and lots of cunning.
The aim is to get your own cards cleared from your half of the deck before your opponent, but the beauty of this game is that you can frequently and utterly maliciously ruin your opponents plans. Hence the name of the game!
Although there is some luck involved as there is in most card games, skill will usually win out and an experieced player will comfortably beat all but the luckiest of beginner.
The game takes a bit of time to get accustomed to so that you can reliably work out how best to dispose of your own cards and how to most effectively block your opponent, but it's very much worth the investment! Once you get into this game there's no going back. The king of 2 player card games!
