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Byo-yomi (秒読み byōyomi?) is an extended time control in two-player games, specifically shogi and go. The word is borrowed from Japanese; the term literally means "counting the seconds," or more generally, "countdown."
A typical time control is "60 minutes + 30 seconds byo-yomi", which means that each player may make as many or as few moves as he chooses during his first 60 minutes of thinking time, but after the hour is exhausted, he must make each move in thirty seconds or less. To enforce byo-yomi, a third person or a game clock with a byo-yomi option is necessary.
In professional Go games and many amateur tournaments, a player has several byo-yomi periods, for example five periods of one minute each. If a player makes his move within a one-minute period, he retains all five periods for his future moves. If a player oversteps one minute, he starts the following move in the second rather than the first byo-yomi period. In effect, the player has one minute per move plus four extra one-minute packets which may be used as needed, e.g. four moves of two minutes each, or one move of five minutes, or any other combination.
In higher-level tournaments, such as the Kisei tournament, the player's time is often composed entirely of byo-yomi periods (for example, in an eight-hour game, the player may have 480 periods of one minute each), rather than having a main block of thinking time. In this case, the actual counting of time (verbally) begins once the player falls below a certain threshold of time, such as 10 minutes; when the time is being counted, the player is informed at intervals how much time they have used in their current period, and how many extra periods they have left. (For example, the time may be called at 10-second intervals, and when 55 and 58 seconds have been used; during a player's final minute, the last 10 seconds are counted one by one.) 
When analog game clocks are used to enforce byo-yomi, it is more convenient to assign additional time for a block of moves, rather than for each move. In Canadian byo-yomi, a player typically gets 5 minutes for 10 to 20 moves. The IGS Go server uses a similar system, but the byo-yomi time is variable and always covers 25 moves. Thus the time control "20 minutes + 15 minutes byoyomi" on IGS means that after the initial 20 minutes of thinking time are over, a player is granted 15 additional minutes, which may be spent however he chooses. If these minutes expire before he has made 25 more moves, he loses. If he makes 25 more moves in less than 15 minutes, he is granted another 15 minutes of byo-yomi, and so on indefinitely.
Canadian byo-yomi imposes a certain average speed of play, but allows to spend more time to ponder on difficult moves. Several byo-yomi periods in one move per period variant (also known as Japanese byo-yomi, though that is a bit of tautology) serve essentially the same purpose, albeit to a lesser extent.
Unused time during one byo-yomi period does not carry forward to future moves. This is in contrast to the Fischer clock often used in chess, with designations such as "5 minutes + 12 seconds per move". Under this time control each player has twelve seconds added to his clock after every move, starting from the first move, regardless of how much time he spends on each move. Thus if a player thinks for eight seconds before making his first move, he will have five minutes and four seconds on his clock after making it.