  We use dense redshift surveys to explore the properties of galaxy clusters
selected from the redMaPPer catalog of overdensities of red galaxies. Our new
survey, HeCS-red (Hectospec Cluster Survey of red-sequence selected clusters),
includes 10,589 new or remeasured redshifts from MMT/Hectospec observations of
redMaPPer clusters at redshifts $z$=0.08-0.25 with large estimated richnesses
(richness estimate $\lambda >64$). Our spectra confirm that each of these
candidate clusters corresponds to an overdensity in redshift space. The
redMaPPer photometric redshifts have a slight bias towards higher redshifts. We
measure the scaling relation between velocity dispersion $\sigma_p$ and
redMaPPer richness estimates $\lambda$. The observed relation shows intrinsic
scatter of 24\% in velocity dispersion at fixed richness, and a range of a
factor of two in measured $\sigma_p$ at fixed richness. We extend our analysis
to HeCS-red-ext, a sample that includes several clusters selected by X-ray flux
or SZ signal. The heterogeneous sample of 121 clusters in HeCS-red-ext shows
similar intrinsic scatter, but the range of $\sigma_p$ at fixed richness
increases to a factor of three. We evaluate the membership probability
estimates $P_{mem}$ for individual galaxies provided by redMaPPer. The
spectroscopic membership fraction is larger than $P_{mem}$ for $0.05\leq
P_{mem}\leq 0.7$; conversely, it is smaller than $P_{mem}$ at $P_{mem}\geq
0.8$. We compare spectroscopic richness estimates to redMaPPer richness
estimates and find good agreement on average, but a range of a factor of two in
spectroscopic richness at fixed redMaPPer richness. Overall, within the
high-richness and low-redshift cut of our sample, spectroscopically estimated
parameters such as velocity dispersion correlate well with photometric richness
estimates, although the relations contain substantial scatter.
