Pointing pairs
The first technique that reaches across the grid. A digit cornered onto one line inside a box lets you erase it two boxes away — without placing anything at all.
The idea
Take one digit and one box. If every square in that box that could hold the digit happens to lie in the same row, then the digit is definitely somewhere in that row, inside this box.
And if it is definitely in that row here, it is definitely not in that row anywhere else — so it can be erased from the six squares of that row in the two neighbouring boxes. The same works with a column.
A worked example
The 5 in row 4 and the 5 in row 6 between them keep the 5 out of the top and bottom lines of the left-middle box. All that is left for it there is row 5 — specifically r5c1 or r5c3.
You still do not know which. But you know the box's 5 is in row 5, so row 5's 5 is in this box, and every other square in row 5 loses it. r5c8 was {1,5}; it is now a 1.
How to scan for it
Go box by box, digit by digit, and ask where the digit can still go. If the answers all share a row or all share a column, you have one. Two or three candidate squares in a line is the shape to look for — hence the names pointing pair and pointing triple.
This and box/line reduction are two halves of one idea, usually taught together as intersection removal: a box and a line overlap in exactly three squares, and information flows both ways across that overlap.
Common mistakes
The error that breaks puzzles is erasing inside the box. The deduction says nothing about the other squares of the box — only about the rest of the line. r5c1 and r5c3 keep all their candidates.
The other is skipping the check that the candidates really are confined. If the digit could also sit in r4c2, there is no pointing pair, and any elimination you make from it is a guess wearing a technique's clothes.
Where to go next
The next technique in the order is box/line reduction — The same overlap argument run the other way — from the line into the box.
All nine techniques: the techniques index. New to the rules? Start here. Ready to use it? Open a board.