Tidiness means not leaving loose ends that could benefit the opponent later. It is not a reaction to a specific imminent threat, but a basic preventative measure that decreases the likelihood of bad things happening in the future. If a player can make a move that carries little cost or risk, but takes away some opportunity from the opponent, it is untidy not to make the move. Making such a move is also called tidying up.
Acute corner example
Consider the following situation, where Red has just played 1 to connect her group to the bottom edge.
Blue would like to play elsewhere on the board. However, this would leave Red with a 2nd row ladder escape along the bottom edge. While this ladder escape may not look immediately threatening to Blue, it would be untidy to just leave it unattended. Instead, Blue tidies up by first playing 2, which forces Red to reconnect, say at 3.
Now Blue has taken away Red's ladder escape and is free to move elsewhere. In fact, Blue also gained a small amount of territory.
To illustrate that this can make a difference, consider the following position, with Blue to move. In this situation, "a" is winning, but "b" and all other moves are losing.
Obtuse corner example
Red has just played 1, threatening to connect to the bottom edge. Blue responds at 2. This will result in a 2nd row ladder along the bottom edge.
Let's assume that Red has a ladder escape, so that Red will "win" the ladder. Red could start the ladder right away, but it is better to first tidy up the corner by playing 3 and 4:
This gains a bit of territory for Red, and often ends up capping a red flank. Note that 3 is immediately forcing: if Blue does not respond, Red connects to the edge. On the other hand, if Red has already played and connected the ladder, 3 is no longer forcing.
To illustrate that this can make a difference, consider the following position, with Red to move. In this situation, "a" is winning, but "b" and all other moves are losing.
Failing to correctly complete a joseki often results in an untidy situation.
In Go, there is the concept of aji, which means something like "the possibilities left in a position".