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.
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