We discuss the measurements of the galaxy cluster mass functions at z approximate to 0.05 and z approximate to 0.5 using high-quality Chandra observations of samples derived from the ROSAT PSPC All-Sky and 400 deg(2) surveys. We provide a full reference for the data analysis procedures, present updated calibration of relations between the total cluster mass and its X-ray indicators (T(X), M(gas), and Y(X)) based on a subsample of low-z relaxed clusters, and present a first measurement of the evolving L(X) - M(tot) relation (with M(tot) estimated from Y(X)) obtained from a well defined statistically complete cluster sample and with appropriate corrections for the Malmquist bias applied. Finally, we present the derived cluster mass functions, estimate the systematic uncertainties in this measurement, and discuss the calculation of the likelihood function. We confidently measure the evolution in the cluster comoving number density at a fixed mass threshold, e.g., by a factor of 5.0 +/- 1.2 at M(500) = 2.5 x 10(14) h(-1) M(circle dot) between z = 0 and 0.5. This evolution reflects the growth of density perturbations, and can be used for the cosmological constraints complementing those from the distance-redshift relation.