sägen
to saw
To cut through or work on something, usually wood, metal, or plastic, with a saw.
Der Tischler sägte das Brett exakt auf die richtige Länge.
The carpenter sawed the board to exactly the right length.

to saw / snore
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to saw
To cut through or work on something, usually wood, metal, or plastic, with a saw.
Der Tischler sägte das Brett exakt auf die richtige Länge.
The carpenter sawed the board to exactly the right length.

In its main meaning, 'sägen' (to saw) is often combined with materials that can be cut. Common objects are 'Holz' (wood), 'ein Brett' (a board), 'ein Ast' (a branch), or 'Metall' (metal).
Er hat den ganzen Nachmittag Holz für den Winter gesägt.
He sawed wood for the winter all afternoon.
The meaning 'laut schnarchen' (to snore loudly) is colloquial. It is used to describe the noise as very loud and disturbing, similar to the sound of a saw. The context is always sleeping.
Mein Hund hat letzte Nacht so laut gesägt, dass ich aufgewacht bin.
My dog snored so loudly last night that I woke up.
The idiom 'an jemandes Stuhl sägen' (to undermine someone's position) is a fixed construction with the preposition 'an' and the dative. It means that someone is trying to secretly weaken another person's position.
Er hat das Gefühl, dass sein neuer Kollege an seinem Stuhl sägt.
He has the feeling that his new colleague is undermining his position.
to cut
'schneiden' and 'sägen' both describe separating or working on material. 'schneiden' is more general and is used with knives, scissors, or other sharp tools. 'sägen' is more specific: you use a saw and usually work on wood, metal, or plastic. Learners may confuse the words because in both cases, part of the material is separated in the end. The basic rule is simple: with a knife or scissors you use 'schneiden'; with a saw you use 'sägen'.

Für das neue Regal sägt Paul ein langes Brett in zwei Teile.
For the new shelf, Paul saws a long board into two pieces.
Am Montag hat Tom ein Stück Stoff für das Kostüm geschnitten.
On Monday, Tom cut a piece of fabric for the costume.