What is intonation and why does it matter?
Intonation is whether your guitar plays in tune with itself as you move up the neck, not whether it's in tune at the headstock. You can tune a guitar perfectly using an open-string tuner and still have it sound wrong the moment you play a barre chord at the seventh fret. That's an intonation problem, and it's a different animal entirely from "my guitar is out of tune."
Why it happens at all
Fretting a string stretches it slightly, sharpening the pitch. The amount of stretch — and therefore the amount of sharpening — varies by string gauge, tension, and how far the string sits above the fret. If the saddle isn't positioned to compensate for this per string, notes fretted higher up the neck will drift sharp compared to the open string, even though your tuner said everything was fine at the nut.
How it's checked and fixed
Compare the pitch of the open string to the same string fretted at the 12th fret. If the fretted note is sharp, the saddle needs to move back (increasing string length) for that string. If it's flat, the saddle needs to move forward. On electrics with individually adjustable saddles, this is a straightforward screwdriver job. On most acoustics, the saddle is a single fixed piece, which means intonation issues sometimes require reshaping or replacing it.
Why you should actually care
If chords sound slightly off, or your guitar sounds "almost" in tune no matter how carefully you tune it, intonation — not your ears, not your tuner — is very often the actual culprit. It's one of the most overlooked parts of a proper setup, and it's the difference between a guitar that sounds right everywhere on the neck and one that only sounds right in first position.