A counter hit is a successful counterattacking hit, when it hits from right after an opponent’s attack hitbox starts up to right after the move ends. Also, “counter hit” refers to the state in a game’s system, if it treats this condition specially.
A counter hit may cause more damage to be dealt (for example, in Street Fighter II, when characters with certain rushing moves are hit while performing them, they will receive much more damage than normal). In nearly all fighting games, a counter hit will deal at least slightly more damage as a reward.
Counter hits often have some graphical effect (if not just a textual notice) displayed, and in some games, it allows one to follow up with a juggle attack.
Sometimes, an attempt at a counter hit will cause both sides to trade hits, but even so, this may end up being advantageous (e.g., causing the opponent to float very high, or to have a longer pushback than you do).
Some games will register a counter hit against normal moves, or even dashes. Most games only register a counter hit against command moves or higher, or special moves or higher, however. Some games also register them against taunts, too.
In KOF, counter hits do not register against normal moves, but they do register against jump attacks from small and medium jumps. Also, some counter hits will cause the opponent to bounce off the sides of the screen in more recent games in the series (“counter wire”). KOF has some moves that do extra damage when they do counter hits (like Ralf’s Galactica Phantom or Shen Woo’s Bakushin); there are also a couple for which the entire point is to score a counter hit (in particular, Ryō’s Tenchi Haōken).