The unknown state of matter at ultra-high density, large proton/neutron number asymmetry, and low temperature is one of the major long-standing problems in modern physics. Neutron stars provide the only known setting in the Universe where matter in this regime can stably exist, and thus serve as natural laboratories for studying the physics of the strong interaction and the state of supranuclear matter. Valuable information about the interior structure of neutron stars can be extracted via sensitive observations of their exteriors using a variety of complementary techniques. This prospect has served as the principal motivation for the Neutron Star Interior Composition Explorer (NICER) X-ray timing observatory, a NASA explorer mission of opportunity that was deployed as an attached payload onboard the International Space Station in June of 2017. The key science objective of NICER is to conduct extensive X-ray timing observations of neutron stars in order to constrain the poorly understood behavior of cold dense matter. We will present the NICER data set of the targets observed for this purpose, describe the technique and models that have been developed by the NICER team to estimate the masses and radii of these neutron stars, and the latest set of results.