getrennt
separate
Not together in terms of space; separated from one another.
Die beiden schlafen in getrennten Zimmern.
The two sleep in separate rooms.

separate / separated
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separate
Not together in terms of space; separated from one another.
Die beiden schlafen in getrennten Zimmern.
The two sleep in separate rooms.

'Getrennt' (separate/separated) is the past participle of the verb 'trennen' (to separate). It is very often used as an adjective to describe a state of separation.
Die beiden Gruppen wurden getrennt voneinander untergebracht.
The two groups were accommodated separately from one another.
If you want to express what something or someone is separated from, you use the preposition 'von' with the dative.
Seit dem Umzug lebt er getrennt von seiner Familie.
Since the move, he has been living separately from his family.
In a restaurant, the question 'Zusammen oder getrennt?' (Together or separately?) is very typical. If you do not want to pay together, you answer 'Getrennt, bitte!' (Separately, please!) or say 'Wir möchten getrennt zahlen.' (We would like to pay separately.)
Kellner: „Zahlen Sie zusammen oder getrennt?“ - Gast: „Getrennt, bitte.“
Waiter: 'Are you paying together or separately?' - Guest: 'Separately, please.'
In the context of couples and relationships, 'getrennt leben' (to live separately) or 'getrennt sein' (to be separated) means that the relationship has ended, but the people are not yet divorced.
Sie sind seit letztem Herbst getrennt und werden sich bald scheiden lassen.
They have been separated since last fall and will soon get divorced.
together
'zusammen' and 'getrennt' form a clear opposite pair. 'zusammen' means that people are together in one place, do something together, or are in a relationship as a couple. 'getrennt' means that people or things are not together, but separate. For couples, 'getrennt' can also mean that the relationship or shared life has ended. When paying, people also say 'getrennt' if each person pays separately.

Im Restaurant bezahlen die beiden Freunde getrennt.
At the restaurant, the two friends pay separately.
Am Wochenende besuchen wir zusammen meine Eltern.
On the weekend, we are visiting my parents together.