Just by way of background, Psalm 51 is attributed to David and is said to have been composed after Nathan confronted David on his illicit relationship with Bathsheba. David had a large number of wives and mistresses but became enthralled with Bathsheba, the then wife Uriah. In a calculated move to gain possession of her, David arranged to have Uriah killed in battle. His sin is less about adultery and more about covetousness, deceit, manipulation, and greed.
Following the confrontation, David went into penance and contrition and it was out of that experience that Psalm 51, Miserere was composed. Miserere is the Latin term for mercy. The beginning of the psalm is "Have mercy on me God in your kindness." I have put the Latin of the text below as Allegri's composition is in Latin. The English version can be found here: Psalm 51.
Miserere mei, Deus, secundum misericordiam tuam;
et secundum multitudinem miserationum tuarum
dele iniquitatem meam.
Amplius lava me ab iniquitate mea
et a peccato meo munda me.
Quoniam iniquitatem meam ego cognosco,
et peccatum meum contra me est semper.
Tibi, tibi soli peccavi et malum coram te feci,
ut iustus inveniaris in sententia tua et aequus in iudicio tuo.
Ecce enim in iniquitate generatus sum,
et in peccato concepit me mater mea.
Ecce enim veritatem in corde dilexisti
et in occulto sapientiam manifestasti mihi.
Asperges me hyssopo, et mundabor;
lavabis me, et super nivem dealbabor.
Audire me facies gaudium et laetitiam,
et exsultabunt ossa, quae contrivisti.
Averte faciem tuam a peccatis meis
et omnes iniquitates meas dele.
Cor mundum crea in me, Deus,
et spiritum firmum innova in visceribus meis.
Ne proicias me a facie tua
et spiritum sanctum tuum ne auferas a me.
Redde mihi laetitiam salutaris tui
et spiritu promptissimo confirma me.
Docebo iniquos vias tuas,
et impii ad te convertentur.
Libera me de sanguinibus, Deus, Deus salutis meae,
et exsultabit lingua mea iustitiam tuam.
Domine, labia mea aperies,
et os meum annuntiabit laudem tuam.
Non enim sacrificio delectaris;
holocaustum, si offeram, non placebit.
Sacrificium Deo spiritus contribulatus;
cor contritum et humiliatum, Deus, non despicies.
Benigne fac, Domine, in bona voluntate tua Sion,
ut aedificentur muri Ierusalem.
Tunc acceptabis sacrificium iustitiae, oblationes et holocausta;
tunc imponent super altare tuum vitulos.