That's American composer Samuel Barber's Agnus Dei, the vocal arrangement of a movement from his String Quartet Op.11 (written in 1936). It eventually became popular as the Adagio for Strings, an arrangement for string orchestra and the key music from the film, Platoon. You're right that the language isn't English (it's Latin). The piece has a distinctly neo-Renaissance sound, with its sacred context, modal idiom and chains of expressive suspensions.
I've always admired Barber (and another American composer, Roger Sessions) for daring to develop personal and cosmopolitan (i.e., non-nationalist) styles. They left the production of musical Americana to Aaron Copland, a twentieth century Jewish composer from Brooklyn who went to Paris to learn how to write cowboy ballets and operas about the Old West.
Perhaps you should listen to the entire quartet. It's possible you'll like it.
Here's a recommendation I'm sure of: look for a CD of the choral music of Samuel Barber (mine is on the ASV label). His Reincarnations, which is on that CD, is very much along the lines of the Agnus Dei, which is also included.
I'm also a fan of Barber's art songs. If you're diligent, you can find on a single budget CD two historic recordings: the Hermit Songs, sung by Leontyne Price with the composer himself on piano, and Barber's Dover Beach, a setting of Matthew Arnold's famous elegy to the Victorian period, which is performed by Dietrich Fischer-Dieskau and the Julliard Quartet.
Still, it sounds as if you're interested in the sound of a capella (fancy schmancy term for unaccompanied) vocal polyphony for a small ensemble, so why not sample the real thing? It doesn't get much better than Monteverdi's "Lasciate mi morire," which can be found in his Madrigals, Book Six (try the Arcana recording by Rinaldo Alessandrini and the Concerto Italiano).
Other possibilities:
A capella:
Josquin: Deploration de la mort de l'Ockeghem
Schnittke: Psalms of Repentance
Gesualdo: Madrigals
Monteverdi: All Madrigals
Thomas Tallis: Spem in Alium (a 40-part motet!); Lamentations of Jeremiah
Ockeghem: Requiem
Vaughan Williams: Mass in G minor
Leonin and Perotin (they were the first)
Machaut: Missa de Notre Dame
Wert
Gombert
Dufay
Schnittke: Requiem
Gorecki: Totus Tuus, op.60 (you'll might like this especially, though there's no real counterpoint)
Brahms's madrigals (yes, he wrote them, and every voice, even the most insignificant, has its own stellar melody)
Accompanied but still relevant:
Monteverdi: Vespro Della Beate Vergine (I like the New London Consort's version.)
Stravinsky: Symphony of Psalms (I sense you'll like the Alleluia)
Bach: Ich Hatte Viel Bekummernis; (Mass in b minor); the chorale, "Est ist genug"
Henze: Cantata della fiaba estrema
Benjamin Britten: War Requiem
Gorecki: Beatus Vir (again, not the most complex piece, but I have the sense you might like it)
Beethoven: Missa Solemnis (let someone else mention Mozart's Requiem, which you probably know already)
(others will say to listen to Arvo Pärt, but I hope you'll listen to real medieval and renaissance music before settling for A.P., whose music is quite pleasing but is a simplistic s-l-o-w-e-d-down version of Stravinsky's take on medieval style)
