A rook ending, both players down to seconds, both banging out moves without looking, the position dead level. One flag falls and the whole game is decided by which player could reach the button faster. That was ordinary tournament chess for most of the last century, and increment is the reason your generation will hardly ever see it.
Increment: seconds you keep
Bobby Fischer patented a clock that adds a fixed amount of time to your clock every time you press it, and the idea was first used in his 1992 rematch with Spassky. It is now the default everywhere, and players call it Fischer increment or just increment.
On 90+30, thirty seconds arrive on your clock each time you complete a move. Over a sixty move game that is thirty extra minutes, handed out in small pieces. It also means you can never reach a position with time to think about and no time to think in: however badly you have managed the first two hours, you still get thirty seconds a move for the rest of your life.
That is why FIDE’s standard control adds the increment from move one rather than only after move forty. The guarantee only works if it is there the whole way.
Delay: seconds you use or lose
Delay works from the other end. With a five second delay your clock does not move for the first five seconds of your turn. Move inside those five seconds and your main time is untouched. Take twelve seconds and you lose seven. Nothing accumulates, ever.
Bronstein delay, named after David Bronstein, produces the same numbers by a different route: your main clock runs from the first second, and when you press, the time you used is credited back up to the delay maximum. Art. 6.3.2 of the FIDE Laws describes this mode as a fixed extra time that has to expire before the main thinking time starts to count down.
Organisers who use delay usually do it for the schedule. A five second delay cannot lengthen a round, while an increment can add half an hour to a long game, and the hall is booked until six.
What it changes about how you play
With thirty seconds a move, technique matters again. Rook endings get played out instead of shuffled, king and pawn races get calculated, and the player who knows how to convert a rook endgame actually gets to convert it.
With a small delay or none at all, the calculation changes. You cannot build a cushion, so you have to arrive at move thirty with real minutes in hand: decide the plan early, spend your time on the two or three moves where the game turns, and play the obvious recaptures instantly.
And in either case, know what happens at zero before it happens to you. Running out of time does not always lose, and the exception has saved me half a point more than once.