"Kosumi" is a Japanese term from the game of Go (Wei Qi/Badouk). In English, it is usually called a diagonal connection. It refers to playing a stone diagonally adjacent to one of your own stones, when there are no other stones in the two points adjacent to both stones. If this is confusing, look at this diagram:

.....
.*,*.
.,o,.
.*,*.
.....

In this picture, White (o) playing at any of the points marked with an asterisk would be kosumi, assuming that the points marked with a comma are empty. If, say, the left comma had a Black stone present, either of the left asterisks would be hane, not kosumi. If it had a White stone present, then a move at one of those points would either be an empty triangle (bad) or a filled triangle (good), depending on whether there was a Black stone at the other relevant comma.

At first, it may seem strange to call this move a connection, since the rules state that two stones of the same color are connected if and only if they are adjacent orthogonally. To understand why we call it a connection, consider this diagram:

....
.ao.
.ob.
....

(No stones at a and b, but see below.)

Imagine that Black (x) decides that he's going to try to cut White's kosumi. He tries playing at a. White can now just play at b, connecting and forming a nice, strong filled triangle in the process. Likewise, if Black plays at b, White plays at a for the same result. Therefore, we say that a and b are "miai" (equivalent points) and, as such, the shape is connected as it stands (although Black may be able to play a or b as kikashi or a ko threat).

Because the stones are connected, but the shape reaches a bit farther than a direct connection (aka a nobi or stretch), kosumi is a very good move when trying to run away from an attack. It covers some ground, increases liberties, and is impossible to cut apart. Another advantage of this over nobi is that the stones are not firmly connected yet, and it is still possible to sacrifice one of them if it is necessary or desireable to do so. Therefore, when a group is running into the center, or towards a stronger friendly group in order to escape attack, the most common moves for the defending player to play are kosumi and ikken tobi.

The problem with kosumi is that it is slow. That is, it doesn't expand influence or territory as much as, say, niken tobi, keima or ogeima. Therefore, it is usually only appropriate in defense, not for attacking or developing.