This might be a tough one. My mom and dad are trying to remember the name of a game they played before I was born (I am 36 so we’re going back). All they can remember is that it was a card game, it had a plastic board with slots that you put the cards into, lying down flat. They can’t remember if they were face up or face down. They think it was sort of a solitaire type game, but for two players. I know this is not a lot to go on. Google is failing me so far.
I agree with comment that the question really needs more details but based on what you've said I will assume your describing Guess Who
It fits your description in following ways...
It came out in 1979 and was certainly around when I was a child and I'm only a little older than you.
The game board had plastic slots in which meant you could have cards standing up or face down.
It was a two player game that was really each player playing their own solitaire game until one of them won.
The only thing I can think of that vaguely matches your description is Rack-o