  We present two new non-parametric methods for quantifying galaxy morphology:
the relative distribution of the galaxy pixel flux values (the Gini coefficient
or G) and the second-order moment of the brightest 20% of the galaxy's flux
(M20). We test the robustness of G and M20 to decreasing signal-to-noise and
spatial resolution, and find that both measures are reliable to within 10% at
average signal-to-noise per pixel greater than 3 and resolutions better than
1000 pc and 500 pc, respectively. We have measured G and M20, as well as
concentration (C), asymmetry (A), and clumpiness (S) in the rest-frame
near-ultraviolet/optical wavelengths for 150 bright local "normal" Hubble type
galaxies (E-Sd) galaxies and 104 0.05 < z < 0.25 ultra-luminous infrared
galaxies (ULIRGs).We find that most local galaxies follow a tight sequence in
G-M20-C, where early-types have high G and C and low M20 and late-type spirals
have lower G and C and higher M20. The majority of ULIRGs lie above the normal
galaxy G-M20 sequence, due to their high G and M20 values. Their high Gini
coefficients arise from very bright nuclei, while the high second-order moments
are produced by multiple nuclei and bright tidal tails. All of these features
are signatures of recent and on-going mergers and interactions. We also find
that in combination with A and S, G is more effective than C at distinguishing
ULIRGs from the "normal" Hubble-types. Finally, we measure the morphologies of
45 1.7 < z < 3.8 galaxies from HST NICMOS observations of the Hubble Deep Field
North. We find that many of the z $\sim$ 2 galaxies possess G and A higher than
expected from degraded images of local elliptical and spiral galaxies, and have
morphologies more like low-redshift single nucleus ULIRGs.
