For example, in Maxwell's theory of electromagnetism, a vacuum solution would represent the electromagnetic field in a region of space where there are no electromagnetic sources (charges and electric currents), i.e. where the current 4-vector vanishes:

$J^a=0$

Another example is Einstein's theory of general relativity where a vacuum solution would represent the gravitational field in a region of spacetime where there are no gravitational sources (masses), i.e. where the energy-momentum tensor vanishes:

$T\_\{ab\}=0$