Vector magnetographs measure the longitudinal (line-of-sight) component of the magnetic field separately from the transverse (image-plane) components, using different aspects of the Zeeman splitting that affects the wavelength of emission and/or absorption spectral lines in the presence of a magnetic field.
The Zeeman splitting is caused by the fact that individual atoms are magnetized due to the circulating motion of electrons bound to them.
Specifically, the circular-polarized component of the light is shifted in wavelength proportional to the field strength in the direction of the observer, and the wavelength shift of the vertical and horizontal linearly-polarized components measures the field strength in those directions.
For example, SOLIS requires about an hour to gather polarized spectral profiles over the whole Sun, and Hinode, the recently launched spacecraft with a 0.5-meter solar telescope on board, takes about an hour to cover a 164-arcsecond-square field (1% of the Sun) at very high spatial resolution.
The splitting effect is antisymmetric along the line-of-sight, but symmetric transverse to the line of sight, so the transverse component of the field can only be measured up to a factor of -1: there is a 180° ambiguity in vector magnetograph measurements of portion of the magnetic field that is perpendicular to the line of sight of the instrument.