die Partie
game
A single game, especially in board games or card games such as chess, skat, or tennis.
Lass uns am Abend eine Partie Schach spielen.
Let's play a game of chess this evening.

game / section
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game
A single game, especially in board games or card games such as chess, skat, or tennis.
Lass uns am Abend eine Partie Schach spielen.
Let's play a game of chess this evening.

'Die Partie' does not mean 'party' in English. A celebration or party in German is 'die Party'. 'Die Partie' usually refers to a game (e.g., chess) or a part of something.
Wir haben am Wochenende eine spannende Partie Tennis gespielt.
We played an exciting game of tennis at the weekend.
The meaning of 'Partie' (game/section) becomes clear from context. In sports or board games, it is a single game. In music, it is the part of an instrument. In trade, it is a shipment of goods.
Die Schachpartie dauerte über vier Stunden.
The chess game lasted over four hours.
In the meaning 'Spiel' (game), 'Partie' is often combined with the verb 'spielen' (to play) and the name of the game. Typical phrases are 'eine Partie Schach spielen' (to play a game of chess), 'eine Partie Karten spielen' (to play a game of cards), or 'eine Partie Billard spielen' (to play a game of billiards/pool).
Wer hat die letzte Partie Poker gewonnen?
Who won the last game of poker?
The expression 'eine gute Partie machen' means to marry someone who is considered advantageous (e.g., because of money or status). The expression is often perceived as somewhat old-fashioned.
Ihre Eltern hofften, dass sie eine gute Partie machen würde.
Her parents hoped that she would make a good match.
When 'Partie' describes a part, it is often combined with another noun to form a compound. Common examples are 'Körperpartie' (part of the body), 'Hautpartie' (area of skin), or 'Gesichtspartie' (area of the face).
Diese Creme ist speziell für die empfindliche Augenpartie entwickelt worden.
This cream was developed especially for the sensitive eye area.
game / play
'das Spiel' and 'die Partie' can both mean a round or a contest with rules. 'Spiel' is the more general word and fits many situations: children play a game, teams have a game or match, friends play a game. 'Partie' usually means one single game, especially in chess, card games, or sometimes tennis. It sounds a bit more precise and often somewhat more formal. The words can be confused because, for example, you can say both 'ein Spiel Schach' and 'eine Partie Schach' (a game of chess). 'Partie' puts more emphasis on the single round.

Nach der Arbeit spielen Mehmet und Paul noch eine Partie Schach.
After work, Mehmet and Paul play another game of chess.
Vor dem Wochenende sucht sie ein Spiel, das man auch zu zweit spielen kann.
Before the weekend, she looks for a game that can also be played by two people.