Extending from a Single Stone

In general, the safe distance to extend from one stone is only one point (a one-point jump). However, because of the edge effect, a stone on the third line can usually support a two-space extension, such as shown in the diagram. Black cannot cut this apart, tactically speaking, without stones around in support.
Not only is this a stable extension, it can also be thought of as the minimum amount of space needed as base on the side to have any hope of creating a living group.

Of course, there are exceptions to the two-space rule.

The diagram here shows an example where a two-point extension should not be made. Making the two-point extension to 1 would mean touching a stone already there. Since Black will probably respond with 2, White will be forced to play 3, which gives her a very flat position. It is clear that this result is not good for her.

However, in general a two-space extension from a single stone on the third line is a good, stabilising move.

Extensions from stones on the fourth line

In general, the reasons for wanting to extend from a stone on the fourth line are not the same as for extending from a stone on the third line. Having played a stone on the fourth line, you have played a stone which aims at influence more than territory (alternatively: more focus on development, less on stability). No single extension from a stone on the fourth line will settle it as nicely as a two-space extension from a third-line stone. As shown in the diagram, extending to a or b leaves the group open to attack at c or d. Even if White extends to c, she can still be attacked by d (or vice versa).

For these reasons, extensions from fourth line stones tend to be longer and less safe, aiming more at influence.

The diagram shows some possible extensions from the corner hoshi point stone:

To a at the hoshi point is the most expansive. The link between these two stones is weak, but the influence of the stones on the bottom left is very large. If this extension is left alone, White can easily construct a large territory here. Sometimes this extension is made on the third line instead.

StephanTerre -- Should this read "the influence of the stones on the upper left"? If not, would it be possible to explain why the stones on the top of the board exert such influence on the bottom of the board? (Sorry if it's a dumb question!)

Charles Matthews Yes, This is a rather roundabout explanation of the framework White has here.

The point b can be played and is actually a safe extension in that it cannot be cut. However, the group is still open at one side, and for example a play by Black on the 3-4 point will take a big chunk out of it. But this is acceptable, White was not playing directly for territory.

However, by playing b on the third line she does indicate that she wants to settle her group to some extent, otherwise she would have played c instead, which is a very influential move, partly so also towards the left side. However, the extension to c is easily cut and invites battles.

Extending to d after playing the hoshi point stone is not consistent with the high position of the hoshi stone. Although it is safer than an extension to b, it suffers the same drawbacks but has a smaller, less ambitious scope.

The extension to e is often played as a respose to a kakari - aiming for influence on the left side, keeping a good connection with the hoshi stone. However, the thus-created group is still open to attack from both sides.

Note that, as was already stated, extending from a corner hoshi stone is not urgent.

Extensions from stones on the fourth line which are not corner hoshi stones can be made as well. However, in almost all of these cases, it is more important to evaluate what you are extending towards than what you are extending from. I.e. the length and height of your extension will depend on the position of your opponent in the direction you are extending. (This is of course true for all extensions.)

Here, a and b are miai, so there is no need for White to make an extension from this stone until Black approaches from one side or the other.