  FRB 121102 is the only known repeating fast radio burst source. Here we
analyze a wide-frequency-range (1-8 GHz) sample of high-signal-to-noise,
coherently dedispersed bursts detected using the Arecibo and Green Bank
telescopes. These bursts reveal complex time-frequency structures that include
sub-bursts with finite bandwidths. The frequency-dependent burst structure
complicates the determination of a dispersion measure (DM); we argue that it is
appropriate to use a DM metric that maximizes frequency-averaged pulse
structure, as opposed to peak signal-to-noise, and find DM = 560.57 +/- 0.07
pc/cc at MJD 57644. After correcting for dispersive delay, we find that the
sub-bursts have characteristic frequencies that typically drift lower at later
times in the total burst envelope. In the 1.1-1.7 GHz band, the ~ 0.5-1-ms
sub-bursts have typical bandwidths ranging from 100-400 MHz, and a
characteristic drift rate of ~ 200 MHz/ms towards lower frequencies. At higher
radio frequencies, the sub-burst bandwidths and drift rate are larger, on
average. While these features could be intrinsic to the burst emission
mechanism, they could also be imparted by propagation effects in the medium
local to the source. Comparison of the burst DMs with previous values in the
literature suggests an increase of Delta(DM) ~ 1-3 pc/cc in 4 years, though
this could be a stochastic variation as opposed to a secular trend. This
implies changes in the local medium or an additional source of
frequency-dependent delay. Overall, the results are consistent with previously
proposed scenarios in which FRB 121102 is embedded in a dense nebula.
