  Some active galactic nuclei (AGN) are surrounded by extended emission-line
regions (EELRs), which trace both the illumination pattern of escaping
radiation and its history over the light-travel time from the AGN to the gas.
From a new set of such EELRs, we present evidence that the AGN in many Seyfert
galaxies undergo luminous episodes 20,000-200,000 years in duration. Motivated
by the discovery of the spectacular nebula known as Hanny's Voorwerp, ionized
by a powerful AGN which has apparently faded dramatically within ~ 100,000
years, Galaxy Zoo volunteers have carried out both targeted and serendipitous
searches for similar emission-line clouds around low-redshift galaxies.We
present the resulting list of candidates and describe spectroscopy identifying
19 galaxies with AGN-ionized regions at projected radii > 10 kpc. This search
recovered known EELRs and identified additional previously unknown cases, one
with detected emission to r = 37 kpc. At least 14/19 are in interacting or
merging systems; tidal tails are a prime source of extraplanar ionized gas. We
see a mix of one- and two-sided structures, with observed cone angles from
23-112 degrees. We consider the energy balance in the ionized clouds, with
lower and upper bounds on ionizing luminosity from recombination and
ionization-parameter arguments, and estimate the luminosity of the core from
the far-infrared data. The implied ratio of ionizing radiation seen by the
clouds to that emitted by the nucleus, for a constant nuclear source, ranges
from 0.02 to > 12; 7/19 exceed unity. Small values imply heavily obscured AGN.
However, large values may require that the AGN has faded over tens of thousands
of years, giving us several examples of systems in which such dramatic
long-period variation has occurred; this is the only current technique for
addressing these timescales in AGN history. (Abridged)
