Finally the time has come to explain the Sudoku Swordfish rule! I've noticed this is one of the most popular search terms leading to this blog, so I'm happy to be able to help out.
Consider this board:
Following elimination, we reach the position below. I've highlighted some areas in red. In row 5, there are two places where we can place a 9, in columns 4 and 6. In row 7, 9 can appear in columns 4 and 9 only, and in row 9 it can appear in columns 6 and 9 only.
The curved red lines show how the cells involved in the swordfish form a kind of loop. Placing a value in one of the cells implies its positions in the rest of the loop. Suppose the 9 in row 5 appears in column 4, that is, in cell (4, 5). Then cell (6, 5) can't contain a 9, so cell (6, 9) must (because it's the only other place in column 6 than can). That implies that (9, 9) can't contain a 9, so (9, 7) must. Alternatively, if the 9 in row 5 appears in cell (6, 5), following things around the loop we see that 9's must also appear in cells (9, 9) and (4, 7). Either way, we've placed a 9 in each of the three columns 4, 6, and 9. We can eliminate any other possible 9's in those columns, namely the ones highlighted in blue.
Another way to see things is to notice is that there are three 9's to be placed in three rows (5, 7, and 9), and exactly three columns (4, 6, and 9) intersecting those rows where they can be placed. Each column must receive a 9 in one of the three rows.
The swordfish can also appear in a vertical orientation. In this case, we notice three columns that each can take a given number only in two places. Between the three columns, the places that can hold that number fall into exactly three rows. The board below provides an example:
I've highlighted the vertical swordfish (on the number 3) in the grid following eliminations:
A swordfish could also involve three rows (or columns) that each have the same three possible places for a number. Say the value is placed in the upper-leftmost cell of the pattern; then the lower-left cell (which lies in the same column) can no longer hold the value. The remaining cells form an X-Wing pattern. The same X-Wing appears if the value belongs in the lower-leftmost cell. I'll try to come up with an example of this type of swordfish in the near future.
So, to summarize, the pattern consists of three rows, and a total of three columns within those rows where a given value can be placed. The value can be removed from the remainder of those columns. The same rule applies if the words "row" and "column" are switched.
I hope this helps to clarify the Swordfish rule!