It is a form for responsorial prayer between a single voice and a whole group: a cantor and the assembly.

The basic form is alternation between verses from a cantor and a repeated response from the assembly. It generally follows the reading. (In the traditional Lauds and Vespers there's a hymn instead.) It is called "brevis" because it's short: the responsories that follow the readings during Vigils/Matins are more wordy ("prolix"). The repeated response is often shorter in the repetitions, with a fuller version (eg verse1+response) only at the start and the end. Sometimes at the start the cantor and the assembly each say the (fuller) response.

Usually the response is the second half of a psalm verse, or perhaps a whole subsequent verse, which "fits" for repetition. So eg at Terce today, the response is Because I have sinned against Thee, and the verses are (1) Heal my soul, (2) I have said, Lord, have mercy on me, and (3) Glory to the Father... as usual as the last one.

You can find music for them for the older Office, in the older books, eg the Antiphonale

As for the Mass: several of the Responsorial Psalms of the modern lectionary have the form of "short responses" (although most of them have rather longer responses, more like antiphons). For example: Trinity Sunday, or the 30th Sunday, in Year A. You could sing those in the brief response style. Or in the Gradual Simplex (By Flowing Waters), the responsorial psalms proposed are very much in the "brief response" style, and use similar chant formulas: but with many verses, not just two or three. So that's another way you could use this style at Mass.

In my experience the faithful may be a little "surprised" if you do, though, because they have to pay attention in a new way in order to participate. They have to sing the response after every verse, for one thing, and not repeat every time exactly the first thing the cantor sang, for another. So it takes some getting used to.