X-Wing
An X-Wing is the simplest of the fish patterns. Pick a digit and look for two rows where that digit's candidates line up in the same two columns. The four corner cells form a rectangle, and because each of the two rows must contain the digit somewhere, the digit has to occupy two opposite corners — one per row, one per column. That locks it out of every other cell in the two columns.
What is an X-Wing?
An X-Wing is a single-digit pattern that spans two rows and two columns at the same time. The setup is: in two different rows, a particular digit can only go in two cells, and those two cells sit in the same two columns in both rows. Draw a line between the four cells and you get a rectangle — that's the "wing". Because each of the two rows has to place the digit once, and the only available spots are inside that rectangle, the digit must land on two opposite corners. One column gets it in the first row, the other column gets it in the second. Either way, both columns now have their digit accounted for, so every other candidate for that digit in those two columns can be erased. The same logic works with the rows and columns swapped.
How to spot it
X-Wings hide in plain sight because they require looking at two rows simultaneously, which most solvers don't do until they're stuck. The reliable way in is to scan by digit: pick a candidate digit and, for each row, note which columns it can land in. If two rows each restrict that digit to exactly two columns, and the two column-pairs match, you've found an X-Wing. The same scan works column-first instead of row-first. X-Wings are uncommon but they turn up regularly in harder puzzles once the singles and subsets have run their course.
The logic
Every row places each digit 1–9 exactly once, and so does every column. If a digit is confined to two cells in each of two rows, and those cells share the same two columns, then the two rows must place the digit in two of the four corner cells — one per row. The two placed digits necessarily fall in different columns (otherwise one column would get two copies of the digit, which is impossible). So each of the two columns receives exactly one digit from the X-Wing, and no other cell in those columns can hold it. The same argument with three rows and three columns gives a Swordfish, and with four, a Jellyfish.