die Partei
party
A political organization of people with similar goals that takes part in elections in order to gain political power.
Er ist Mitglied in einer großen politischen Partei.
He is a member of a large political party.

party / party to
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party
A political organization of people with similar goals that takes part in elections in order to gain political power.
Er ist Mitglied in einer großen politischen Partei.
He is a member of a large political party.

Careful: 'die Partei' never means a celebration or social event. A party is 'die Party' or 'die Feier' in German. 'Die Partei' is usually a political organization.
Am Samstag gehe ich auf eine Party, nicht zu einer Partei.
On Saturday I'm going to a party, not to a political party.
In a political context, 'Partei' (party) is often used with verbs like 'wählen' (to vote for), 'gründen' (to found), or 'einer Partei beitreten' (to join a party) / 'in einer Partei sein' (to be in a party).
Viele junge Leute wollen einer politischen Partei beitreten.
Many young people want to join a political party.
The meaning 'Vertragspartner' (contracting party) or 'Prozessbeteiligter' (party to proceedings) is technical language. You can recognize this meaning through signal words like 'Vertrag' (contract), 'Gericht' (court), 'Anwalt' (lawyer), or 'Konflikt' (conflict).
Beide Parteien müssen den Vertrag unterschreiben.
Both parties must sign the contract.
The fixed expression is 'Partei ergreifen für' + accusative (to take sides with / in favor of). It means that you side with someone in an argument and support that person.
In dem Streit ergriff er Partei für seinen besten Freund.
In the argument, he took sides with his best friend.
When 'Partei' is mentioned in connection with a board game or card game, it means a single round or a single game.
Wollen wir noch eine Partei Schach spielen, bevor du gehst?
Do we want to play another game of chess before you leave?
page / side
'die Seite' and 'die Partei' can both refer to people or groups in a dispute, contract, or conflict. 'Seite' is more general here: you stand on someone's side, or one side has an opinion. 'Partei' is more specific when it is about a person or group involved in a legal dispute, contract, or conflict. In the phrase 'Partei ergreifen' (to take sides), it means something similar to: to place yourself on one side. Learners may confuse the words because both occur in conflicts. For a position in an argument, 'Seite' often sounds more natural; for official or legal participants, 'Partei' fits better.

Vor Gericht musste jede Partei ihre Unterlagen vorlegen.
In court, each party had to present its documents.
In der Diskussion stellte sich Maria auf die Seite ihrer Schwester.
In the discussion, Maria took her sister's side.