  The extraplanar diffuse ionized gas (eDIG) in 17 nearby, edge-on disk
galaxies is studied using deep Taurus Tunable Filter Halpha and [N II] lambda
6583 images and conventional interference filter Halpha + [N II] lambda 6548,
6583 images which reach flux levels generally below about 1 x 10^{-17} erg
s^{-1} cm^{-2} arcsec^{-2}. [N II] lambda 6583/Halpha excitation maps are
available for 10 of these objects. All but one galaxy in the sample exhibit
eDIG. The contribution of the eDIG to the total Halpha luminosity is relatively
constant, of order 12 +/- 4%. The Halpha scale height of the eDIG derived from
a two-exponential fit to the vertical emission profile ranges from 0.4 kpc to
17.9 kpc, with an average of 4.3 kpc. This average value is noticeably larger
than the eDIG scale height measured in our Galaxy and other galaxies. This
difference in scale height is probably due in part to the lower flux limits of
our observations. A quantitative analysis of the topology of the eDIG confirms
that several galaxies in the sample have a highly structured eDIG morphology.
The distribution of the eDIG emission is often correlated with the locations of
the H II regions in the disk. A strong correlation is found between the star
formation rate (IR luminosity) per unit disk area and the extraplanar ionized
mass, providing further support for a strong connection between the disk and
eDIG components in these galaxies. The excitation maps confirm that the [N
II]/Halpha ratios are systematically higher in the eDIG than in the disk. More
detail is given in Paper II. (abridged)
