der Springer
jumper
A person who jumps, especially as an athlete in disciplines such as track and field, ski jumping, or diving.
Der Skispringer landete bei einer Weite von 140 Metern.
The ski jumper landed at a distance of 140 meters.

jumper / knight
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jumper
A person who jumps, especially as an athlete in disciplines such as track and field, ski jumping, or diving.
Der Skispringer landete bei einer Weite von 140 Metern.
The ski jumper landed at a distance of 140 meters.

The word 'Springer' (jumper) is derived directly from the verb 'springen' (to jump). The ending '-er' refers to the person or thing that performs the action. This pattern is very common in German, e.g. 'laufen' (to run) → 'der Läufer' (the runner).
Der Springer bereitete sich auf den Absprung vor.
The jumper prepared for the takeoff.
You can recognize the meaning of 'Springer' (jumper / knight / floater) from the context. In sports, it refers to athletes; in chess, to the piece; and in the office, to a flexible employee.
Der Wasserspringer gewann die Goldmedaille.
The diver won the gold medal.
In sports, 'Springer' (jumper) is usually used as part of a compound word that names the discipline. Examples are 'Skispringer' (ski jumper), 'Wasserspringer' (diver), or 'Weitspringer' (long jumper).
Der Skispringer landete sicher auf beiden Beinen.
The ski jumper landed safely on both feet.
In the workplace, a 'Springer' (floater) is a person without a fixed position who is assigned wherever staff are missing. The matching verb is 'einspringen' (to fill in).
Unsere Abteilung ist froh, dass morgen ein Springer für den kranken Kollegen einspringt.
Our department is glad that tomorrow a floater will fill in for the sick colleague.
horse
'das Pferd' and 'der Springer' can refer to the same piece in chess. In everyday life, people first think of the animal when they hear 'Pferd', but in chess 'Pferd' can be used colloquially for the piece. 'Springer' is the more precise and usual term in chess. The words can be confusing because 'Pferd' has several meanings and 'Springer' can also be a person who jumps or steps in flexibly. When it is clearly about chess, both are understandable, but 'Springer' sounds more precise.

Max hat mit dem Springer die Dame angegriffen.
Max attacked the queen with the knight.
Beim Schach setzt Lena ihr Pferd in die Mitte des Brettes.
In chess, Lena moves her knight to the center of the board.