A wagering requirement (also known as playthrough or rollover) is a mathematical multiplier that dictates how many times you must bet the bonus funds before they turn into withdrawable cash. For example, if you have a bonus and the requirement is a specific multiplier, you must wager that amount multiplied by the bonus value in total bets.
Why does this exist? Bonuses are promotional tools. To prevent players from simply claiming a bonus and immediately withdrawing it, operators implement these requirements to ensure the funds are used for actual gaming. You will often see two types of requirements: Bonus-only (where only the rewarded credit is wagered) and Deposit+Bonus (where both your deposit and the reward are combined into the total requirement). The latter results in a higher total amount to bet but often comes with a lower multiplier.
Progress tracking is usually handled automatically by the system. You can monitor your remaining "playthrough" in your account settings. The exact multiplier for any specific offer is always displayed clearly alongside the promotion before you click "Claim."