Almost any card game is good for brain exercise, but games that make you think or plan, such as bridge, whist and poker, are particularly beneficial. Research is showing that people who remain active mentally, live longer and avoid or delay brain disorders, such as dementia.
Many card games have a competitive style/activity or a social style. Either style is important, and you can choose the level of competition you want.
Bridge offers a competitive form called Duplicate Bridge, and a social form called Contract (Party)Bridge. Many people who try bridge and other more competitive games, stop playing after awhile, because other people took the game too seriously. This can be a problem. No game should be taken too seriously, and if you find yourself in a club that does have these types of people, you can probably find other clubs that are more friendly.
Poker is now very popular, and although it is mainly based on luck, a good grasp of mathematics helps the luck, and in most forms of poker, a good assessment of the other players also helps the luck! Blackjack requires memorizing a strategy to improve winning odds, so it is excellent too.
Other games that don't use playing cards, can be useful for brain exercise too. Domino games, such as Mexican Train, are fun but do involve some planning, which is good for the brain. Many commercial games such as Skip Bo, offer alternatives to regular playing cards.
Games are mainly social activities and that is good too. Socializing is also good for mental health. Some experts suggest that mental health is just as important as physical health for the elderly. Certainly a combination of both, is the best for you.
When you can't play socially, solitaire card games do offer some brain exercise too.