An expression of regret at having caused trouble for someone.

Roget 950: penitence, contrition, compunction, repentance, remorse; regret etc. 833.    self-reproach, self-reproof, self-accusation, self-condemnation, self-humiliation; stings of conscience, pangs of conscience, qualms of conscience, prickings of conscience^†, twinge of conscience, twitch of conscience, touch of conscience, voice of conscience; compunctious visitings of nature^†.    acknowledgment, confession etc. (disclosure) 529; apology etc. 952; recantation etc. 607; penance etc. 952; resipiscence^†.    awakened conscience, deathbed repentance, locus paenitentiae [Lat.], stool of repentance, cuttystool^†.    penitent, repentant, Magdalen, prodigal son, a sadder and a wiser man" [Coleridge]. ... show more

Roget 952: atonement, reparation; compromise, composition; compensation etc. 30; quittance, quits; expiation, redemption, reclamation, conciliation, propitiation; indemnification, redress.    amends, apology, amende honorable^†, satisfaction; peace offering, sin offering, burnt offering; scapegoat, sacrifice.    penance, fasting, maceration, sackcloth and ashes, white sheet, shrift, flagellation, lustration^†; purgation, purgatory.    ... show more

Dutch: excuus, verontschuldiging, spijtbetuiging
Polish: przeprosiny

A formal written defense of something you believe in strongly.

synonym: apologia.

A poor example:
— It was an apology for a meal.

synonym: excuse.

Roget 937: vindication, justification, warrant; exoneration, exculpation; acquittal etc. 970; whitewashing.    extenuation; palliation, palliative; softening, mitigation.    reply, defense; recrimination etc. 938.    apology, gloss, varnish; plea etc. 617; salvo; excuse, extenuating circumstances; allowance, allowance to be made; locus paenitentiae [Lat.].    apologist, vindicator, justifier; defendant etc. 938.    justifiable charge, true bill.    v.. justify, warrant; be an excuse for etc. n.; lend a color, furnish a handle; vindicate; exculpate, disculpate^†; acquit etc. 970; clear, set right, exonerate, whitewash; clear the skirts of.    extenuate, palliate, excuse, soften, apologize, varnish, slur, gloze; put a gloss, put a good face upon; mince; gloss over, bolster up, help a lame dog over a stile.    advocate, defend, plead one's cause; stand up for, stick up for, speak up for; contend for, speak for; bear out, keep in countenance, support; plead &c., 617; say in defense; plead ignorance; confess and avoid, propugn^†, put in a good word for.    take the will for the deed, make allowance for, give credit for, do justice to; give one his due, give the Devil his due.    make good; prove the truth of, prove one's case; be justified by the event.    ... show more