die Heizung
heating system
The technical system or device used to heat rooms, buildings, or water.
Die Heizung ist ausgefallen, deshalb ist es in der Wohnung eiskalt.
The heating system has broken down, so the apartment is ice-cold.

heating
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heating system
The technical system or device used to heat rooms, buildings, or water.
Die Heizung ist ausgefallen, deshalb ist es in der Wohnung eiskalt.
The heating system has broken down, so the apartment is ice-cold.

'Die Heizung' (heating/heating system) is derived from the verb 'heizen' (to heat). The ending '-ung' forms a feminine noun that can refer both to the device or system (meaning 1) and to the process of heating (meaning 2).
Die Heizung des gesamten Gebäudes ist sehr modern.
The heating system for the entire building is very modern.
In everyday life, 'Heizung' (heating system) is often combined with certain verbs: you can 'anmachen' or 'einschalten' (turn it on), 'ausmachen' or 'ausschalten' (turn it off). To regulate the temperature, you say 'aufdrehen' (turn up, warmer) or 'runterdrehen' (turn down, colder).
Kannst du bitte die Heizung im Wohnzimmer etwas aufdrehen?
Could you please turn up the heating in the living room a bit?
When a heating system is not working, there are typical phrases. For example, you say: 'Die Heizung ist ausgefallen' (the heating has broken down), 'Die Heizung ist kaputt' (the heating is broken), or 'Die Heizung wird nicht warm' (the heating isn't getting warm).
Im Winter ist es besonders schlimm, wenn die Heizung ausfällt.
In winter, it is especially bad when the heating breaks down.
warmth / heat
'die Wärme' and 'die Heizung' both have to do with warm rooms. 'Wärme' means what you feel: it is not cold, but pleasant or hot. It can come from the sun, from the body, or from a heating system. 'Heizung' usually means the device or system that produces heat, for example in a room or in a house. The words can be confused because a 'Heizung' produces 'Wärme'. In practical terms: you turn on the 'Heizung', but you feel the 'Wärme'.

Anna hat bei ihren Eltern die Heizung höher gestellt, weil es im Gästezimmer kalt war.
At her parents' house, Anna turned up the heating because it was cold in the guest room.
Nach dem langen Weg durch den Schnee tat mir die Wärme im Wohnzimmer gut.
After the long walk through the snow, the warmth in the living room felt good.