Third line capture
The third line capture is one of the basic edge patterns.
Non-cutting stone
In the first place, if the third line stone is not cutting, there is no urgent need to capture it. Why did the opponent leave it in danger? Probably because there are larger plays elsewhere on the board. That said....
Here we see a generic side situation. For White to capture black immediately (that is, without giving it more than two liberties), she must place a stone that interferes with Black's escape.
anon: What's wrong with giving Black more than two liberties?
xela: It depends on the context. Sometimes, there's nothing wrong with it. But often, if black has three liberties, then there's an opportunity to push through and cut somewhere and start attacking the surrounding stones.
I propose this as a killing sequence without extra support.
All of the marked white stones in the diagram above make it possible to capture black. There are many such combinations; these are just a few examples.
The marked spots are sente against the Black stone : a White move at any of these spots threatens to capture Black immediately.
(A White move at some of these spots is sometimes called the Maeda method, after Maeda Ryo, who taught it in the US. But it is not clear exactly what he taught... the only sources so far are hearsay. There is a video by tuerda (4d North American amateur) which discusses this position and gives many applications.)
The marked spots are also effective depending on the ladder status.
Even without supporting stones, White may try something like this for outside influence, which is often good for White ( may be at a or b, or at driving Black to the other direction). In such cases, Black may well abandon the stone.
Cutting stone
When the stone is cutting, there are fewer effective supports. White may choose the "outside influence" sequence above.
On the other hand, Black may play for outside influence himself by sacrificing the stone. This is often seen in practice. may be at a or b.
A related tesuji in joseki
This is a shape that occurs in a common joseki. White needs to protect the cutting point at a somehow.
However, if white plays here, black takes an excellent shape point, creating a crocodile's mouth, and white cannot be satisfied. The same will occur if white protects the cutting point with either of the tiger's mouth plays (marked a). White should extend.
is a joseki move; x and y instead of are also joseki. As you can see, after the exchange at a, at b, matches a marked spot in the "sente moves" diagram above.
See also Bishop's net for a related shape.