die Runde
group
A group of people gathered for a conversation, a game, or a social activity.
In gemütlicher Runde saßen wir noch lange zusammen.
We sat together for a long time in a cozy group.

round
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group
A group of people gathered for a conversation, a game, or a social activity.
In gemütlicher Runde saßen wir noch lange zusammen.
We sat together for a long time in a cozy group.

The noun 'die Runde' (the round) is derived from the adjective 'rund' (round). The basic idea of a circle or circular movement connects all the meanings: a circle of people, a round in sports, a round of drinks for everyone in the circle.
Wir saßen in einem runden Kreis um das Lagerfeuer.
We sat in a round circle around the campfire.
When 'Runde' (group) means a group of people, it often appears with the preposition 'in'. Typical expressions are 'in gemütlicher Runde' (in a cozy group), 'in großer Runde' (in a large group), or 'in die Runde fragen' (to ask the group).
Wir haben den Abend in kleiner Runde ausklingen lassen.
We wound down the evening in a small group.
In the context of sports or competitions, fixed expressions like 'eine Runde weiterkommen' (to advance one round), 'die nächste Runde erreichen' (to reach the next round), or 'in der ersten Runde ausscheiden' (to be eliminated in the first round) are very common.
Die Mannschaft ist eine Runde weitergekommen.
The team advanced one round.
The expression 'eine Runde ausgeben' or 'eine Runde schmeißen' means that you order and pay for a drink for everyone present in a group.
Weil er Geburtstag hat, gibt er heute eine Runde aus.
Because it's his birthday, he's buying a round today.
The fixed expression 'die Runde machen' (to make the rounds) means that information spreads quickly. The subject of the sentence is the information, e.g., a rumor, a piece of news, or a joke.
Das Gerücht über den neuen Chef machte schnell die Runde.
The rumor about the new boss quickly made the rounds.
When a person walks an inspection route, the expression 'seine Runde machen' or 'seine Runde drehen' is often used. This is typical for professions such as night watchmen or security staff.
Der Polizist dreht seine Runde durch das Viertel.
The police officer makes his round through the neighborhood.
society / company
'die Gesellschaft' and 'die Runde' can both mean a group of people who are together. 'Gesellschaft' tends to emphasize being together or the fact that you are not alone. 'Runde' is especially fitting when several people are sitting together, talking, playing, or doing something together. It often sounds more concrete and more casual. The words can be confused when talking about a small group. 'In Gesellschaft' means more like 'with other people'; 'in der Runde' refers more to the specific group there on the spot.

In der kleinen Runde erzählte jeder kurz von seinem Wochenende.
In the small group, everyone briefly talked about their weekend.
Im Urlaub genießt Paul die Gesellschaft seiner Freunde.
On vacation, Paul enjoys the company of his friends.