There's two ways to do it:
The first is to run two threads; one waits for input and updates state accordingly while the other runs the game.
The other (more common in game development) way is to write the game as one big loop that executes many times a second, updating game state, redrawing the screen, and checking for input.
But instead of blocking when you get key input, you check for the presence of pending keypresses, and if nothing has happened, you just continue through your loop. If you have multiple input sources (keyboard, network, etc.) they all get put there in the loop, checking one after another.
Yes, it's called polling. No, it's not efficient. But high-end games are usually all about pulling the maximum performance and framerates out of the computer, not running cool.
For added efficiency, you can optionally block with a timeout -- saying "wait for a keypress, but no longer than 300 milliseconds" so you can continue on with your loop.
select()
comes to mind, but there are other ways of waiting or checking for input as well.