der Mitspieler
teammate
A person who takes part in the same game, sport, or activity, often on the same team.
Er passte den Ball zu seinem Mitspieler vor dem Tor.
He passed the ball to his teammate in front of the goal.

teammate / fellow player
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teammate
A person who takes part in the same game, sport, or activity, often on the same team.
Er passte den Ball zu seinem Mitspieler vor dem Tor.
He passed the ball to his teammate in front of the goal.

The word is made up of the preposition 'mit' (with) and the noun 'Spieler' (player). So it literally means a person who 'mitspielt' (plays along/participates). This structure is very logical and easy to understand.
Im Team ist es wichtig, gut mit seinen Mitspielern zu kommunizieren.
In a team, it is important to communicate well with your teammates.
The prefix 'Mit-' always expresses community or doing something together. You find it in many words such as 'Mitschüler' (classmate) or 'Mitbewohner' (roommate/flatmate).
Genau wie ein „Mitschüler“ mit dir lernt, spielt ein „Mitspieler“ mit dir im selben Team.
Just as a 'Mitschüler' (classmate) learns with you, a 'Mitspieler' (teammate) plays with you on the same team.
In business or political contexts, 'Mitspieler' (player) is used for companies or actors that are active in a particular field. It is a metaphor that presents the market as a kind of playing field.
Apple und Google sind wichtige Mitspieler auf dem Smartphone-Markt.
Apple and Google are important players in the smartphone market.
player
'der Spieler' and 'der Mitspieler' can both mean a person who is taking part in a game, sport, or activity. 'Spieler' is more general: a person plays, for example in a team, in a club, or in a game. 'Mitspieler' emphasizes shared participation with others more strongly. A 'Mitspieler' is therefore someone who plays together with me or with a group. Learners may confuse the words because both fit games and sports. If only the role as a person who plays is important, 'Spieler' fits. If the relationship to the other participants is important, 'Mitspieler' sounds more natural.

Nach dem Training wartet Max auf seine Mitspieler aus dem Team.
After training, Max waits for his teammates from the team.
Der neue Spieler hat im ersten Spiel gleich ein Tor geschossen.
The new player scored a goal right away in his first game.