  Active galactic nuclei (AGNs) with double-peaked [O III] lines are suspected
to be sub-kpc or kpc-scale binary AGNs. However, pure gas kinematics can
produce the same double-peaked line profile in spatially integrated spectra.
Here we combine integral-field spectroscopy and high-resolution imaging of 42
double-peaked [O III] AGNs from Sloan Digital Sky Survey to investigate the
constituents of the population. We find two binary AGNs where the
line-splitting is driven by the orbital motion of the merging nuclei. Such
objects account for only ~2% of the double-peaked AGNs. Almost all (~98%) of
the double-peaked AGNs were selected because of gas kinematics; and half of
those show spatially resolved narrow-line regions that extend 4-20 kpc from the
nuclei. Serendipitously, we find two spectrally unresolved binary AGNs where
gas kinematics produced the double-peaked [O III] lines. The relatively
frequent serendipitous discoveries indicate that only ~1% of binary AGNs would
appear double-peaked in Sloan spectra and 2.2_{-0.8}^{+2.5}% of all Sloan AGNs
are binary AGNs. Therefore, the double-peaked sample does not offer much
advantage over any other AGN samples in finding binary AGNs. The binary AGN
fraction implies an elevated AGN duty cycle (8_{-3}^{+8}%), suggesting galaxy
interactions enhance nuclear accretion. We illustrate that integral-field
spectroscopy is crucial for identifying binary AGNs: several objects previously
classified as "binary AGNs" with long-slit spectra are most likely single AGNs
with extended narrow-line regions. The formation of extended narrow-line
regions driven by radiation pressure is also discussed.
