Hang fire refers to an unexpected delay between the triggering of a firearm and the ignition of the propellant. This failure was common in firearm actions that relied on open primer pans, due to the poor or inconsistent quality of the powder. Modern firearms are susceptible also. The delay is typically too brief to be noticed, but can be several seconds.
A hangfire should be suspected whenever a firearm fails to fire, but has not clearly malfunctioned. In modern firearms, it is more likely that the round is a "dud" (one that will never fire at all), but it is important to delay removing the round from the chamber. If a hangfire has in fact occurred, a round initating outside of the firearm could cause serious injury. If the operator believes that the firearm correctly cycled a fresh round and the action is not visibly jammed, the correct procedure is to keep the firearm pointed at a safe target for thirty seconds, then remove the round. This rule is usually not followed in combat, where being without a working firearm is the bigger risk.
The phrase "to hang fire" also means to delay in progressing, for example from one task to another.