We present an analysis of 23 L dwarfs whose optical spectra display unusual features. Twenty-one were uncovered during our search for nearby, late-type objects using the Two Micron All-Sky Survey while two were identified in the literature. The unusual spectral features, notably weak FeH molecular absorption and weak Na I and K I doublets, are attributable to low-gravity and indicate that these L dwarfs are young, low-mass brown dwarfs. We use these data to expand the spectral classification scheme for L0 to L5-type dwarfs to include three gravity classes. Most of the low-gravity L dwarfs have southerly declinations and distance estimates within 60 pc. Their implied youth, on-sky distribution, and distances suggest that they are members of nearby, intermediate-age (~10-100 Myr), loose associations such as the Beta Pictoris moving group, the Tucana/Horologium association, and the AB Doradus moving group. At an age of 30 Myr and with effective temperatures from 1500 to 2400 K, evolutionary models predict masses of 11-30 M_Jupiter for these objects. One object, 2M 0355+11, with J-K_s=2.52+/-0.03, is the reddest L dwarf found in the field and its late spectral type and spectral features indicative of a very low gravity suggest it might also be the lowest-mass field L dwarf. However, before ages and masses can be confidently adopted for any of these low-gravity L dwarfs, additional kinematic observations are needed to confirm cluster membership. ; Comment: Accepted to AJ