It's a term used in computer music for unwanted delay. 'Unwanted delay' sounds like a bad thing, which it is, so manufacturers disguise it by using a word that few people understand.
Audio cables have no latency. Analog audio equipment - mixing consoles, tape recorders etc. - has no latency. Digital equipment does have latency. Computers have it by the bucketful.
Latency causes two main problems...
One is that there is a delay in the straight-through path between input and output. This would cause a problem if you were recording a vocal to an already-recorded backing track. The singer would hear himself or herself slightly delayed in the headphones. This is incredibly distracting.
The cure in latency-prone systems is to provide a special monitor output which takes the signal from BEFORE the point it enters the computer (or recorder). The disadvantage is that it is usually desirable to EQ, compress and add reverb to the monitor signal. EQ and compression are not possible, and the delay might adversely affect any reverb.
The other problem is that you might record one track, then subsequent overdubs don't play back exactly in sync. In this case, whole tracks must be shifted in time to compensate.
The best solution is to use a system whose latency is so small that it is hardly noticeable.
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