  The oxygen abundance gradients among nebular emission line regions in spiral
galaxies have been used as important constraints for models of chemical
evolution. We present the largest ever full-wavelength optical spectroscopic
sample of emission line nebulae in a spiral galaxy (M31). We have collected
spectra of 253 HII regions and 407 planetary nebulae with the Hectospec
multi-fiber spectrograph of the MMT. We measure the line-of-sight extinction
for 199 HII regions and 333 PNe; we derive oxygen abundance directly, based on
the electron temperature, for 51 PNe; and we use strong line methods to
estimate oxygen abundance for 192 HII regions and nitrogen abundance for 52 HII
regions. The relatively shallow oxygen abundance gradient of the more extended
HII regions in our sample is generally in agreement with the result of Zaritsky
et al. (1994), based on only 19 M31 HII regions, but varies with the
strong-line diagnostic employed. Our large sample size demonstrates that there
is significant intrinsic scatter around this abundance gradient, as much as 3
times the systematic uncertainty in the strong line diagnostics. The intrinsic
scatter is similar in the nitrogen abundances, although the gradient is
significantly steeper. On small scales (deprojected distance < 0.5 kpc), HII
regions exhibit local variations in oxygen abundance that are larger than 0.3
dex in 33% of neighboring pairs. We do not identify a significant oxygen
abundance gradient among PNe, but we do find a significant gradient in the [N
II] ratio that varies systematically with surface brightness. Our results
underscore the complex and inhomogeneous nature of the ISM of M31, and our
dataset illustrates systematic effects relevant to future studies of the
metallicity gradients in nearby spiral galaxies.
