Astrometry is the branch of astronomy for measuring the position of celestial bodies such as stars. I've also seen it described as the precise measurement of bodies on the celestial sphere. The first definition includes some resulting analysis/interpretation. Angular resolution of telescopes used for astrometry plays a role in the resulting precision, but through analysis of multiple observations, positions of stars can be refined as much as down to 1/10 the "official" angular resolution of the instrument.
I've seen the phrases absolute astrometry and relative astrometry, which I interpret to mean determining an exact position in the celestial sphere, versus determining a position relative to other objects in the celestial sphere. The most precise astrometry requires analysis of the errors, i.e., pushing the precision to the maximum that observation and statistical analysis of the noise can provide. The term global astrometry refers to efforts to do this using data collected throughout the celestial sphere, and adjusting the frame of reference and each coordinate so as to be consistent with the most likely noise distribution. This is generally done with space telescope observations and a numerical analysis procedure that includes the hundreds of thousands of individual objects observed. Modern high-precision astrometry of stars is dominated by special-purpose telescopes, not by any means the largest, but on space missions: Hipparcos circa 1990 and Gaia recently, which have measured positions of over a billion stars to unprecedented precision.
Precise measurement of the apparent motion of a star in the sky can reveal its distance through parallax, or can reflect actual movement, showing something of its motion vector, or can indicate an astrometric binary or the presence of an extra-solar planet. Astrometry is of interest in star catalogs used for navigation and for aiming telescopes (i.e., using a guide star).
The terms spherical astronomy and positional astronomy refer to the general science of determining the position of objects in the celestial sphere at different times, including where the Sun, Moon, and planets will be. The term astrometry, as a "measurement", is sometimes used to mean the measurement portion of this science.