der Karton
box
A box or container made of cardboard, usually used for packaging, transport, or storage.
Für den Umzug müssen wir noch Kartons besorgen.
We still need to get boxes for the move.

box / cardboard
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box
A box or container made of cardboard, usually used for packaging, transport, or storage.
Für den Umzug müssen wir noch Kartons besorgen.
We still need to get boxes for the move.

The meaning depends on whether the word is used countably or uncountably. 'Ein Karton' (a box) is a countable container, while 'aus Karton' (made of cardboard) describes the uncountable material.
Ich brauche einen großen Karton für diese Bücher.
I need a large box for these books.
In the context of containers, 'Karton' (box) is often associated with moving or storage. Typical words and phrases are 'Umzugskarton' (moving box), 'Kartons packen' (to pack boxes), or 'Kartons stapeln' (to stack boxes).
Alle Bücher sind schon in Kartons verpackt.
All the books are already packed in boxes.
When 'Karton' means the material cardboard, it often appears with the preposition 'aus' (made of). In this use, it has no plural.
Die Verpackung ist aus recyceltem Karton.
The packaging is made of recycled cardboard.
As a packaging unit, 'ein Karton' (a carton) refers almost exclusively to a large pack containing several cartons or packs of cigarettes.
Er hat am Flughafen einen ganzen Karton Zigaretten gekauft.
He bought a whole carton of cigarettes at the airport.
paper
'das Papier' and 'der Karton' both have to do with material used for writing, printing, or packaging. 'Papier' is thin and flexible; it is used, for example, for letters, notebooks, or printouts. 'Karton' can mean the thicker, firmer material, but also the box made from it. The words can be confused because 'Karton' is made from a paper-like material. A practical rule: for a thin sheet, say 'Papier'; for sturdy packaging or a box, say 'Karton'.

Julia packt die Bücher in einen stabilen Karton.
Julia packs the books into a sturdy box.
Bei der Arbeit schreibt sie die Telefonnummer auf ein kleines Stück Papier.
At work, she writes the phone number on a small piece of paper.