The Ko rule (Basic rule 8) prohibits to make a move that results in a repetition of position (stone allocation) on the entire board.  This rule serves to prevent the possibility of endless recaptures that might otherwise tie the game.

It is important to note that what's prohibited by the Ko rule (Basic rule 8) is a repetition of position on the entire board, not just stone recapture at the point of the original capture.  The following sequence is a legitimate one (though senseless — such positions are normally left as-is until game ends — White is "dead" unconditionally here).

Thus the Ko rule permits Black in fact to make two successive moves in the local position (capture and connection).  This is a great advantage considering that the fate of large groups of stones might depend on the ability to make two successive moves within or at the borders of these groups.  The disadvantage is that the opponent also gets the opportunity to make two successive moves in some other local position.  So if opponent wants to prevent the player from taking an advantage of two moves in a row in one local position, he looks all over the board for another local position where he can create a threat of making two successive moves, and causing the damage that overweights the advantage the first player might get in the first local position (a Ko-threat).

These opportunities of playing two moves in a row locally lead to so-called Ko-fights in Go, which can change dramatically the balance of the entire game, turning the lost game into a victory with only several moves, and visa versa.

White group seems to be "alive" in the left position below, considering White's threat to cut and capture black stone by making a move to the point X.  This threat calls Black to protect his weak point X, which gives White an opportunity to play at point Z and "live" unconditionally.  Exchange of Black 1 and White 2 is one possibility of such peaceful development.

Let's suppose, however, that in some other locality Black has a group of an unconditionally "dead" stones shown in the left position below.  There are only two ways make a "live" group with two "eyes" out of this position.  Both ways require that Black will be able to make two successive moves in this locality without White's resistance.  Hope that White won't notice Black's first approaching move is quite naive.  However, the the Ko-fight initiated elsewhere might prevent White from answering to Black's approaching move in this position for the possibility of a bigger loss in another local position.

Left position above is Black's Ko-threat that Black can utilize at the right time in order to win a Ko-fight elsewhere on the board.

Black's gain of 14 points is definitely more than the potential black gain from connecting at point X in initial position (which leads also to loosing the chance to save black group in another corner).  Black's gain from destroying white group in initial position might be even more, but before trying to do this Black must find at least one Ko-threat big enough to prevent White from connecting with move 6.  Evaluation of subsequent potential Ko-threats on both sides is also important for a prolonged Ko-fight in order to be reasonably sure that one can win it.

If Black will try to make two "eyes" in the corner without initiating a Ko-fight first, just hoping that White won't pay due attention to Black's approaching move, Black most likely will be stopped by White and will loose both the Ko-threat and potential opportunity to save his group.  Such a play, though not prohibited by Go rules, shows not only a disrespect of the opponent, but also a lack of understanding of the possibilities of Ko-fight.

Similar to the threats of occupying own "critical" points in an attempt to save otherwise "dead" group are the threats of occupying "critical" points within opponent's otherwise "live" group.  What matters most in Ko-fight is the balance between loss and gain, i.e., finding the adequate Ko-threats — just enough to force the opponent to make certain move in certain local position.  It hardly makes any sense to use a large threat in the situation when the smaller threat will give the same result — the habit to assess potential Ko-threats, sorting them according to potential gain or loss, might prove to be very helpful in the cause of the game.

Since potential White loss of their stones in the corner is obviously greater than their loss due to letting Black live in another corner, the most likely outcome of the presented position is White connecting and saving his stones.  In exchange for that opportunity White has to permit Black to make two moves in a row in another corner making a "live" group with two "eyes".

Situations discussed above are intentionally simplistic in order to make the idea of Ko-fight as clear as possible.  The real Ko-fight usually goes all over the board often with several local positions involved where it is difficult to evaluate on the fly the gain-and-loss balance for each position.  It makes sense to take a good note of all potential Ko-threats and never waste them.  A plenty of Ko-threats permits a player to be invulnerable in the Ko-fight, or to be able to launch one himself.  In a well-balanced game the outcome of a Ko-fight can easily determine the result of the entire game.