
die Trauer
grief
A deep feeling of pain and sorrow, usually triggered by the death of someone close or another serious loss.
Nach dem Tod ihres Hundes war sie von tiefer Trauer erfüllt.
After the death of her dog, she was filled with deep grief.
grief / mourning
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grief
A deep feeling of pain and sorrow, usually triggered by the death of someone close or another serious loss.
Nach dem Tod ihres Hundes war sie von tiefer Trauer erfüllt.
After the death of her dog, she was filled with deep grief.

mourning
The period of mourning after a death, often associated with certain social rituals such as wearing dark clothing (mourning clothes).
In manchen Kulturen ist es üblich, ein ganzes Jahr lang Trauer zu tragen.
In some cultures, it is customary to wear mourning for an entire year.
'Die Trauer' (grief/mourning) is the noun related to the verb 'trauern' (to grieve/mourn) and the adjective 'traurig' (sad). All of these words describe the feeling of great pain and loss.
Er ist sehr traurig, weil sein Freund weggezogen ist.
He is very sad because his friend moved away.
You can often recognize the meaning 'period of mourning' from verbs like 'tragen' (to wear/bear, in the sense of wearing mourning clothes) or from time expressions like 'Trauerjahr' (year of mourning). Without this context, the feeling is usually meant.
Ihre Trauer war für alle sichtbar.
Her grief was visible to everyone.
To indicate the reason for grief, German uses 'über' (+ accusative) for an event or 'um' (+ accusative) for a lost person or thing.
Ihre Trauer über den Verlust ihres Arbeitsplatzes war groß.
Her grief over the loss of her job was great.
Common expressions are 'in tiefer Trauer sein' (to be in deep grief) or 'seine Trauer verarbeiten' (to process one's grief). The feeling is often described as 'tief' (deep) or 'groß' (great).
Es braucht Zeit, die Trauer nach einem schweren Verlust zu verarbeiten.
It takes time to process the grief after a serious loss.
suffering
„Das Leid“ is a general term for great physical or emotional pain and can have many causes. „Die Trauer“ more specifically refers to emotional pain after a death or another serious loss.

Nach dem Tod seiner Frau war seine Trauer sehr groß.
After his wife's death, his grief was very deep.
Die lange Krankheit und der Tod seiner Frau brachten ihm großes Leid.
His wife's long illness and death caused him great suffering.