die Tonne
bin
A large container for waste, usually made of plastic or metal.
Kannst du bitte den Müll in die Tonne bringen?
Can you please take the rubbish to the bin?

bin / ton
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bin
A large container for waste, usually made of plastic or metal.
Kannst du bitte den Müll in die Tonne bringen?
Can you please take the rubbish to the bin?

The meaning of 'Tonne' depends heavily on the context. Words like 'Müll' (rubbish/garbage) point to the waste container, numbers and 'wiegen' (to weigh) to the unit of weight, 'Regenwasser' (rainwater) to a barrel, and 'Schiff' (ship) to a nautical marker.
Der LKW transportiert zehn Tonnen Sand.
The truck is transporting ten tons of sand.
In everyday life in Germany, specific bins for separating waste are often mentioned. Typical ones are 'die Papiertonne' (paper bin), 'die Biotonne' (organic waste bin), and 'die Restmülltonne' (general waste bin).
Denk daran, morgen wird die Papiertonne geleert.
Remember, the paper bin will be emptied tomorrow.
As a unit of measurement (1,000 kg), 'Tonne' often appears in the singular after a number, even though the plural would be correct. The phrase 'Tonnen von etwas' means colloquially 'a lot of something'.
Dieser Stein wiegt über eine Tonne.
This stone weighs over one ton.
The expression 'etwas in die Tonne treten' (or 'klopfen', 'werfen') means that an idea, plan, or object is completely unusable and can be discarded.
Diese Idee ist so kompliziert, die kannst du in die Tonne treten.
This idea is so complicated that you can throw it in the bin.
In the meaning 'Boje' (buoy), 'Tonne' is a technical term from shipping. You can recognize this meaning from words like 'Hafen' (harbor), 'Fahrwasser' (fairway/channel), 'navigieren' (to navigate), or 'Kapitän' (captain).
Das Schiff muss zwischen den grünen und roten Tonnen bleiben.
The ship must stay between the green and red buoys.
barrel
'die Tonne' and 'das Fass' can both be large containers. The difference is in their typical use: a 'Tonne' is often a bin for waste or a simple large container made of plastic or metal. A 'Fass' is usually rounded or bulging and is used especially for liquids such as wine, beer, or oil. The words can be confused because both can be round and large. For waste, you almost always say 'Tonne'; for stored drinks or oil, 'Fass' sounds more natural.

Nach dem Aufräumen stellt Paul die volle Tonne an die Straße.
After tidying up, Paul puts the full bin out by the street.
Im Keller lagert ein großes Fass mit altem Wein.
A large barrel of old wine is stored in the cellar.