As Wikipedia already details, there are many different draughts variants, but there are two common rules and one exotic rule according to which kings can capture.
Long-ranged kings. This is the most common rule, which applies to e.g. the International/Brazilian/Canadian, Russian, Spanish and Czech draughts variants and to Pool checkers in the USA. A king can "scan" across empty squares for capture victims, and it can also "slide" through empty squares directly behind a jumped piece. Each of these potential "landing" squares can be used as the starting point for "scanning" for the next victim, which can be in a direction orthogonal to the current direction.
Short-ranged kings. This applies to American checkers and Italian draughts. A king can only capture a piece when it is directly adjacent to it, and it has to "land" directly behind a jumped piece from which it can continue jumping other pieces, including in a direction orthogonal to the current direction.
Exotic kings. The prime example is Thai draughts, where the king can "scan" across empty squares for capture victims, but where it has to "land" directly behind a jumped piece, and can only "scan" for its next victim from that same landing square.
For almost all draughts variants, the jumped pieces are only removed from the board when the king is on its destination square. Furthermore, it is not allowed to jump the same piece more than once during a capture sequence. Changing directions in a capture sequence is therefore only by 90 degrees left or right.
For Thai draughts, however, each jumped victim is immediately removed from the board, after which the king can continue capturing more victims, including in the direction it came from (i.e. 180 degrees turned).