Live time controls fall into two general categories: move-based and sudden death.
Both types of time control can also include an increment, which is some number of seconds added to the player's clock after each move. For example a 2-second increment would increase a player's clock by two seconds each time the player completes a move. Generally speaking the purpose of an increment is to provide some extra thinking time in games which involve long endgames, and allow games to end by checkate rather than time scrambles where each player simply tries to move faster than the opponent.
Time controls are notated by specifying the moves required (for move-based controls) or the letter "G" (for sudden-death controls), followed by a slash, then the number of minutes, a plus sign, and the increment. For example "30/60+0" means 30 moves in 60 minutes with no increment, while "G/3+2" means the entire game in 3 minutes with a 2-second increment after each move.
Corr time controls specify a certain number of moves which must be completed within a certain number of days. Once the required number of moves has been completed, another time period begins with the same parameters. Any extra time left from the first period is carried over to the next period. The standard time control for corr games is 20/30, which means that 20 moves must be completed within 30 days, followed by another 20 moves in an additional 30 days, etc.