A failure or breakdown (of a machine, device, etc.); something being out of order.
エアコンが故障して部屋が暑くなった。
The air conditioner broke down and the room became hot.
朝、電車が故障して遅延が出ている。
This morning a train broke down and there are delays.
コンピューターの故障でデータが消えてしまった。
A computer failure caused the data to be lost.
An injury or damage to a part of the body (often used in contexts like athletes injuring a limb).
選手は試合中に腰を故障して欠場した。
The player injured his back during the match and was absent.
痛みが治らないので、膝の故障で手術を受けることになった。
The pain wouldn't go away, so he had to have surgery for his knee injury.
A problem, hitch, or hindrance (sometimes used more broadly as an obstacle or impediment).
手続き上の小さな故障があって、申請が遅れている。
There was a small procedural hitch, and the application is delayed.
システムの故障が原因で納期に支障が出た。
A system problem caused delays in the delivery schedule.
Objection or protest (archaic/literary usage; not common in modern everyday speech).
その判決に対して当事者は故障を申し立てた（古い表現）。
The party lodged an objection to the ruling (archaic expression).