der Läufer
runner
A person who runs, especially as a sport or leisure activity.
Beim diesjährigen Marathon waren über zehntausend Läufer am Start.
At this year's marathon, over ten thousand runners were at the starting line.

runner / runner rug / bishop
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runner
A person who runs, especially as a sport or leisure activity.
Beim diesjährigen Marathon waren über zehntausend Läufer am Start.
At this year's marathon, over ten thousand runners were at the starting line.

All meanings of 'Läufer' derive from the verb 'laufen' (to run/walk). The ending '-er' indicates a person or thing that performs the action of the verb. The basic idea is always movement along a line or surface.
Die Kinder laufen schnell durch den Park.
The children run quickly through the park.
The meaning can only be understood from context. If it is about sports, it means a person. If it is about the home, such as a hallway or stairs, it means a carpet. In the context of chess, it is a game piece, and in the garden it is a plant shoot.
Der Läufer überquerte als Erster die Ziellinie.
The runner crossed the finish line first.
When a person is meant, 'Läufer' (runner) is often connected with sports or places. Typical combinations are 'Marathon-Läufer' (marathon runner), 'ein schneller Läufer' (a fast runner), or 'die Läufer am Start' (the runners at the starting line).
Am Sonntagmorgen treffen sich viele Läufer im Stadtpark.
On Sunday morning, many runners meet in the city park.
In the meaning of 'Teppich' (carpet), the word is often used with verbs such as 'legen' (to lay), 'auslegen' (to lay out), or 'reinigen' (to clean). Adjectives often describe the color or material, e.g. 'ein roter Läufer' (a red carpet / runner rug).
Für die Feier wurde ein roter Läufer ausgerollt.
A red carpet was rolled out for the celebration.
bishop
'Bischof' and 'Läufer' can mean the same piece in chess: the piece that moves diagonally across the board. In German, 'Läufer' is the normal word for this. 'Bischof' is more likely to appear when translating from English or talking about international chess terms. The words can therefore be confusing: in everyday German, 'Bischof' usually means a high-ranking church official, not a chess piece. In chess, 'Läufer' is almost always the more natural choice.

Am Montag hat Tom im Schachclub seinen Läufer auf die lange Diagonale gestellt.
On Monday, Tom placed his bishop on the long diagonal at the chess club.
In der englischen Schachnotation entspricht der Bischof dem deutschen Läufer.
In English chess notation, the bishop corresponds to the German Läufer.