The dissertation focuses on the study of rotation-powered pulsars, the primary observational manifestation of neutron stars. These objects are powerful sources of electromagnetic radiation and relativistic particles whose emission is provided by the loss of pulsar rotational energy. Understanding the evolution of pulsars, which happens over billion year timescales, requires detection and study of pulsars at different stages of evolution. I present detailed X-ray analyses of pulsars at four distinct stages of evolution and compare their emission behavior with that of other pulsars expected to be in similar evolutionary stages. I also show key characteristics of the pulsars that make them unique in their group. I start with a young and energetic pulsar, PSR J2022+3842 (characteristic age approximately 9 kyr, spin-down power = 3E37 erg/s), with powerful non-thermal emission. X-ray timing of the pulsar revealed double-peaked X-ray profile with a period twice the previously established value. Our analysis allowed us to update the pulsar's spin-down power and X-ray efficiency using the correct timing results, which brought the pulsar more in-line with other young X-ray pulsars. I also provide the phase-dependent behavior of the pulsar's non-thermal emission. Pulsars with true ages, often substituted by characteristic age, below ~ 100 kyr are considered young and ones with ages > 1 Myr are considered old, with the 'middle-aged' pulsars in the middle. My next pulsar is a 1.8 Myr old J1836+5925 (spin-down power = 1E34 erg/s), which is perhaps the brightest X-ray source among the oldest pulsars still observable in the gamma-rays. Detailed timing and spectral analyses show strong evidence of an absorption feature (perhaps an electron cyclotron line) in the pulsar's spectrum. Characterizing its thermal emission might have important implications for the neutron star cooling models. Moving another two orders of magnitude up in characteristic age, we have one of the oldest known non-recycled X-ray pulsars, PSR J0108-3430, with characteristic age = 166 Myr and spin-down power = 5.8E30 erg/s. The pulsar's spectrum likely consists of a thermal component, emitted from a hot polar cap, and a non-thermal component, emitted from its magnetosphere. The X-ray pulse profile shows a single, asymmetric peak which could be explained by an axially-asymmetric temperature distribution at the pole or by the non-thermal emission from the outer gap. The three pulsars represent important stages in the evolutionary path that a hypothetical single young pulsar like J2022+3842 might take, as it passes through stages close to gamma-ray emission turn-off (like J1836+5925) and X-ray turn-off (similar to J0108-3430).Pulsars in binaries can follow an alternative path. By accreting matter from their companions they can be 'recycled' to short millisecond periods and emit X-rays and gamma-rays for billions of years. I also present a special class of such recycled pulsars which are believed to be in the process of fatally ablating their companions. I present the X-ray analysis of PSR J1446-4701, a pulsar with spin-down power = 3.6E34 erg/s in a 6.7 hr binary orbit, and PSR J1311-3430, a pulsar with spin-down power = 4.9E34 erg/s, in an extreme 1.6 hr binary orbit. PSR J1446-4701 turned out to be a non-eclipser with possibly low (face-on) orbital inclination, with emission from both the pulsar and the intra-binary shock observable throughout the binary orbit. PSR J1311-3430 is a known eclipser, in which we find hints of spectral variability between pulsar superior and inferior conjunction phases. I also present a comprehensive comparison of the sample of such extremely low-mass binary pulsars. We reveal the true nature of pulsars, slowly and steadily, usually one target at a time, but eventually we expect useful patterns to emerge that improves our understanding of the population of rotation powered pulsars.