loneliness / solitude
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loneliness
The feeling of being alone and separated from others, often connected with sadness or a feeling of abandonment.
Nach dem Tod ihres Mannes litt sie sehr unter der Einsamkeit.
After her husband's death, she suffered greatly from loneliness.
'Einsamkeit' (loneliness/solitude) is a noun derived from the adjective 'einsam' (lonely). The ending '-keit' is often used in German to form abstract nouns from adjectives that describe a state or quality.
Er fühlte sich einsam, also litt er unter Einsamkeit.
He felt lonely, so he suffered from loneliness.
Whether the negative or positive meaning is intended can be recognized from context. Verbs like 'leiden unter' (to suffer from) or 'fürchten' (to fear) point to the negative feeling, while verbs like 'genießen' (to enjoy), 'suchen' (to seek), or 'wählen' (to choose) refer to desired seclusion.
Viele alte Menschen leiden unter Einsamkeit.
Many elderly people suffer from loneliness.
In connection with the negative feeling, 'Einsamkeit' (loneliness) is often used with the preposition 'unter' (under/from), as in 'unter Einsamkeit leiden' (to suffer from loneliness). Expressions like 'die Einsamkeit bekämpfen' (to fight loneliness) or 'Angst vor der Einsamkeit haben' (to be afraid of loneliness) are also typical.
Er hat Angst vor der Einsamkeit im Alter.
He is afraid of loneliness in old age.
society / company
'die Gesellschaft' and 'die Einsamkeit' are opposites when it comes to closeness to other people. Here, 'Gesellschaft' means that you are together with others or have their company. 'Einsamkeit' means that you are alone or feel separated from others. It can be sad, but it can also be calm and deliberately chosen. So if you 'Gesellschaft sucht' (seek company), you do not want to be alone.
Auf der langen Reise spürte Tom zum ersten Mal echte Einsamkeit.
On the long journey, Tom felt real loneliness for the first time.
Nach dem Umzug freute sich Maria über die Gesellschaft ihrer neuen Nachbarn.
After moving, Maria was happy to have the company of her new neighbors.