In the recording industry, an active monitor is a monitor (a neutral speaker) that contains its own power source and amplifier.
Every recording studio uses monitors. Their aim is to provide neutral field-level playback of a recording to the engineers and producers, so they can accurately tweak and refine the production without worrying that any imperfections they hear are a result of the studio playback system.
Most monitors are passive monitors - their only input is speaker wire, and they require an external amplifier and powered source in order to work - hence, passive. These have the advantage of being very versatile, but are limited in their reliance on other pieces of the audio puzzle to function properly. Many a passive monitor has been regarded as "junky" through no fault of its own; it was just paired with poor audio equipment and reflected this poverty in its ever-passive manner.
An active monitor, by containing its own power source and amplifier, is the portable response to a passive monitor. This makes it ideal for PA speakers in live settings, where excess processing and amplifying equipment (and the wiring required to manage them) are untenable. It also has the advantage of being able to power down independently, which can eliminate it from any troubleshooting of the audio system quickly unlike a passive monitor, which can only be "muted" - a less than perfect failsafe.
Excellent producers of active monitors include Mackie, KRK, and Synthax.
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