What you are referring to is the precision with which the astrometry (position) of stars will be measured with the Gaia satellite.
The sentences you refer to in Perryman et al. (2014) are "Gaia will achieve accuracies of some 10μas (microarcsec) in positions and annual
proper motions for bright stars (V ∼ 10), degrading to around 25μas at V = 15, and to around 0.3 mas (300μas) at V = 20 (Lindegren et al. 2008)."
What this means is that after the 5-year Gaia mission is complete, the angular position (or coordinates) of 10th magnitude stars on the sky will be determined with a relative precision of 10 millionths of an arcsecond. This will also be the accuracy with which the annual tangential motion (the so-called proper motion) of stars can be measured. i.e. 10th magnitude stars will have an uncertainty in their proper motion of $pm 10 mu$as/year.
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