A solar mass is a unit sometimes used in astronomy or astrophysics to give the mass of objects. A solar mass is equal to the mass of the sun, which is, according to the NASA fact sheet, 1.989 * 10^30 kilograms. Which I would give in plain English, besides there is really no common way to describe that number, because the phrase "2,000 octillion tons" doesn't really give much of a sense of scale for most people.
The main use of the solar mass is to describe the relative sizes of stars, with the mass of our familiar one used as a yardstick. For example, the minimum mass that a star can usually sustain nuclear fusion is given as .10 solar masses, and the minimum mass of a star to form a black hole is somewhere around 3 to 5 solar masses. Objects as small as planets and as large as galaxies can also be described in terms of solar masses, but it usually used to describe the masses of stars and similar objects.