The Y-Wing (also called XY-Wing) brings together three cells with two candidates each. One is the pivot: it has two numbers, for example the 4 and the 5. The other two are the pincers (or wings) and they see the pivot: one has the 4 and a third number, the 7; the other has the 5 and the same 7. Whatever the pivot's number, one of the pincers has to be the 7. So a cell that sees both pincers cannot be 7.
It is the first technique that reasons by cases: «if the pivot is 4…, if it is 5…». This is not guessing. Both cases are followed through to the end and reach the same conclusion, which holds whichever case is true.
An example
There are no singles, locked candidates, subsets or X-Wings in this grid. The cell in row 8, column 3 has only two candidates: the 4 and the 5. It will be the pivot.
In its box, the cell in row 9, column 1 has the 4 and the 7. In its column, the cell in row 5 has the 5 and the 7. These are the pincers: each shares a number with the pivot, and both share the 7.
If the pivot is 4, the pincer in the box cannot be 4 and is left with the 7. If the pivot is 5, it is the pincer in the column that is left with the 7. Either way, one of the two pincers is 7.
The cell in row 5, column 1 sees both pincers: the one in the box through column 1, the one in column 3 through row 5. Whichever pincer has the 7, this cell cannot be 7, and the 7 is removed from its notes.
The cells that see both pincers are where their units cross: in the row of one and the column of the other, or in the box of one and the row or column of the other. That is where you look for the third number. The pivot doesn't count: it also sees both pincers, but it doesn't have the third number.
What is not a Y-Wing
Pincers with different third numbers. 4 and 7 in one, 5 and 8 in the other: no number is guaranteed in either of them.
A pincer with the same two numbers as the pivot. If the pivot and a pincer both have the 4 and the 5, they are a naked pair, not a Y-Wing.
A pincer that doesn't see the pivot. Each pincer has to be in a row, column or box of the pivot. The pincers don't need to see each other.
Cells with three candidates. All three cells must have exactly two. With a third number in the pivot, reasoning by cases no longer covers every possibility.
How to search
Start with the cells that have two candidates. The three cells of the pattern have two each. In a hard grid, they are the first place to look.
For each possible pivot, look for pincers in its units: one with one of the pivot's numbers and a third, another with the other number and the same third.
See which cells see both pincers. If any of them has the third number in its notes, that number is removed.
Trust the notes. The Y-Wing depends on cells with exactly two candidates. An extra note hides the pattern; a missing one invents a pattern that isn't there. See pencil marks.
Your turn
Find a Y-Wing and cross out the third number in the cells that see both pincers. Tap each cell, choose the number and, at the end, press «Check».