  We analyze 24 binary radio pulsars in the North American Nanohertz
Observatory for Gravitational Waves (NANOGrav) nine-year data set. We make
fourteen significant measurements of Shapiro delay, including new detections in
four pulsar-binary systems (PSRs J0613$-$0200, J2017+0603, J2302+4442, and
J2317+1439), and derive estimates of the binary-component masses and orbital
inclination for these MSP-binary systems. We find a wide range of binary pulsar
masses, with values as low as $m_{\rm p} =
1.18^{+0.10}_{-0.09}\text{M}_{\odot}$ for PSR J1918$-$0642 and as high as
$m_{\rm p} = 1.928^{+0.017}_{-0.017}\text{M}_{\odot}$ for PSR J1614$-$2230
(both 68.3\% credibility). We make an improved measurement of the Shapiro
timing delay in the PSR J1918$-$0642 and J2043+1711 systems, measuring the
pulsar mass in the latter system to be $m_{\rm p} =
1.41^{+0.21}_{-0.18}\text{M}_{\odot}$ (68.3\% credibility) for the first time.
We measure secular variations of one or more orbital elements in many systems,
and use these measurements to further constrain our estimates of the pulsar and
companion masses whenever possible. In particular, we used the observed Shapiro
delay and periastron advance due to relativistic gravity in the PSR J1903+0327
system to derive a pulsar mass of $m_{\rm p} =
1.65^{+0.02}_{-0.02}\text{M}_{\odot}$ (68.3\% credibility). We discuss the
implications that our mass measurements have on the overall neutron-star mass
distribution, and on the "mass/orbital-period" correlation due to extended mass
transfer.
