die Begegnung
encounter
A usually accidental meeting with one or more people.
Die Begegnung mit meinem alten Schulfreund auf der Straße hat mich sehr gefreut.
The encounter with my old school friend on the street made me very happy.

encounter / match
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encounter
A usually accidental meeting with one or more people.
Die Begegnung mit meinem alten Schulfreund auf der Straße hat mich sehr gefreut.
The encounter with my old school friend on the street made me very happy.

'Die Begegnung' (encounter/match) comes from the verb 'begegnen' (to encounter/meet), which always requires the dative. This grammar often carries over to the noun: people speak of 'Begegnung mit jemandem' (encounter with someone, dative).
Ich begegne meinem Nachbarn jeden Morgen.
I meet my neighbor every morning.
Whether it means a chance meeting or a sports contest is easy to recognize from context. If the sentence is about people in everyday life, it is an encounter. If it is about sports, teams, or a competition, it is a match.
Es war eine rein zufällige Begegnung am Bahnhof.
It was a purely chance encounter at the train station.
When it is about a meeting, 'Begegnung' (encounter) is often described with adjectives like 'zufällig' (chance/accidental), 'unerwartet' (unexpected), 'kurz' (brief), 'flüchtig' (fleeting), or 'unvergesslich' (unforgettable).
Es war nur eine flüchtige Begegnung, aber ich erinnere mich an ihr Lächeln.
It was only a fleeting encounter, but I remember her smile.
In a sports context, the structure 'die Begegnung zwischen [Team A] und [Team B]' (the match between [Team A] and [Team B]) is almost always used to name the two opponents.
Millionen von Zuschauern verfolgten die Begegnung zwischen den beiden Rivalen.
Millions of viewers watched the match between the two rivals.
game / play
'das Spiel' and 'die Begegnung' can both mean a contest between two teams or people in sports. 'Spiel' is the normal and very common word in everyday language. 'Begegnung' sounds more like sports reporting, newspapers, or official language. Learners may confuse the words because both can refer to the same contest. But it is important to know that 'Begegnung' also means a meeting between people, while 'Spiel' in sports means the contest itself. In conversation, people usually say 'Spiel'; in reports, 'Begegnung' is also common.

Die Begegnung zwischen den beiden Mannschaften war bis zur letzten Minute spannend.
The match between the two teams was exciting until the last minute.
Das Spiel gegen die Nachbarstadt beginnt am Samstag um drei Uhr.
The game against the neighboring town starts on Saturday at three o'clock.