When your opponents stones are diagonally connected, he can be cut.
Here is a pretty common position (result of the 3x3 invasion joseki). Here white can be cut by black at D18 and black can be cut by white at E16. Note that white can't cut black at C15 due to the Tiger's Mouth black created.
Cutting helps you split groups apart and limits their ability to connect with other groups and share liberties. In the position above, black has 2 groups with 1 eye each. If he can connect at P3, the groups will live. If white cuts there, it is VERY unlikely the black groups will survive (although I do see a potential cut for black at R6).
Going back to the first diagram - what if white tries to cut here? That first white stone has no other help on this side of the wall. If black plays at F15, he can quickly contain and kill the cutting white group. So how do you successfully use the cut?
This is the point of this post! Here white approaches to prepare to cut black. To have a better chance at success, white set up the cut from a distance. It is easy for a beginner to completely miss the intent of this move. Study this diagram! Burn the image into your mind - this is a strong attack on black! You want to see this every time your opponent does it in a game. Memorize the danger of the cut!
If Black misses the intent of this and tries to surround white with something like G15, white can spring the trap and cut white with E16! Now black can not really stop white - the cut is successful. Black can try to get white in atari with E15 or F16, but the position has miai for white - whichever spot black takes, white can take the other to connect and escape.
So, pay close attention around spots where you have not solidly connected - diagonals are dangerous! A play that looks like a simple territory grab or a low threat surrounding move may be setting up a cut. At the beginning, I didn't see anything scary in moves that didn't touch my stones. Now I am starting to see these threats that are a row away from my groups - hopefully this post will help you see threats like this more easily as well.
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