Etymology
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Words related to ad-

*ad- 
Proto-Indo-European root meaning "to, near, at."

It forms all or part of: abate; ado; ad-; ad hoc; ad lib; adage; adagio; add; adjective; adore; adorn; adult; adverb; advertise; agree; aid; alloy; ally; amontillado; amount; assure; at; atone; exaggerate; paramount; rapport; twit.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit adhi "near;" Latin ad "to, toward;" Old English æt.
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ap- (1)
form of Latin ad- in compounds with words or stems beginning in -p-; see ad-. In Old French reduced to a-, but scribal re-doubling of ap- to app- in imitation of Latin began 14c. in France, 15c. in England, and was extended to some compounds formed in Old French or Middle English that never had a Latin original (appoint, appall).

In words from Greek, ap- is the form of apo before a vowel (see apo-).
accursed (adj.)
also accurst, early 13c., acursede "being under a curse," past-participle adjective from obsolete verb acursen "pronounce a curse upon, excommunicate" (late 12c.), from a- intensive prefix (see a- (1)) + cursein "to curse" (see curse (v.)). The unetymological -c- is 15c., a mistaken Latinism in imitation of words in acc-. Weakened sense of "worthy of a curse, damnable" is from 1590s. Related: Accursedly; accursedness.
afford (v.)
Middle English aforth, from Old English geforðian "to put forth, contribute; further, advance; carry out, accomplish," from ge- completive prefix (which in Middle English regularly reduces to a-; see a- (1)) + forðian "to further," from forð "forward, onward" (see forth).

The prefix shift to af- took place 16c. under mistaken belief that it was a Latin word in ad-; change of -th- to -d- took place late 16c. (and also transformed burthen, spither, murther, etc. into their modern forms).

The notion of "accomplish" (late Old English) gradually became "be able to bear the expense of, have enough money" to do something (late 14c.), and the original senses became obsolete. Of things, "be capable of yielding," 1580s, which is the sense in afford (one) an opportunity. Related: Afforded; affording.
adjourn (v.)
Origin and meaning of adjourn
mid-14c., ajournen, "assign a day, fix a day" (for convening or reconvening of an organized body), from Old French ajorner (12c.) "meet" (at an appointed time), from the phrase à jorn "to another day, to a (stated) day," from à "to" (see ad-) + journ "day," from Latin diurnus "daily," from dies "day," from PIE root *dyeu- "to shine."

The notion is of setting a date for re-meeting. Meaning "to close a meeting" (with or without intention to reconvene) is from early 15c. Meaning "to go in a body to another place" (1640s) is colloquial. The English word has had the -d- since 16c., but the spelling is unetymological, as the compound apparently is not from Latin; Middle French also occasionally has adjourner, but this was rejected in Modern French. Related: Adjourned; adjourning.
advance (v.)
mid-13c., avauncen (transitive), "improve (something), further the development of," from Old French avancir, avancier "move forward, go forward, set forward" (12c., Modern French avancer), from Vulgar Latin *abanteare (source of Italian avanzare, Spanish avanzar), from Late Latin abante "from before," composed of ab "from" (see ab-) + ante "before, in front of, against" (from PIE root *ant- "front, forehead"). Compare avant.

The unetymological -d- was inserted 16c. on mistaken notion that initial syllable was from Latin ad-. From c. 1300 as "to promote, raise to a higher rank." Intransitive sense "move forward, move further in front" is mid-14c.; transitive sense "bring forward in place, move (something) forward" is from c. 1500. Meaning "to give (money, etc.) before it is legally due" is first attested 1670s. Related: Advanced; advancing. The adjective (in advance warning, etc.) is recorded from 1843.
address (v.)
early 14c., "to guide, aim, or direct," from Old French adrecier "go straight toward; straighten, set right; point, direct" (13c.), from Vulgar Latin *addirectiare "make straight" (source also of Spanish aderezar, Italian addirizzare), from ad "to" (see ad-) + *directiare "make straight," from Latin directus "straight, direct" past participle of dirigere "set straight," from dis- "apart" (see dis-) + regere "to direct, to guide, keep straight" (from PIE root *reg- "move in a straight line"). Compare dress (v.)).

Oldest sense in English is preserved in golf (to address a ball). Meaning "direct for transmission, write as a destination on a message" is from mid-15c. Meaning "to direct spoken words (to someone)" is from late 15c. Late 14c. as "to set in order, repair, correct." The attempt (falsely) re-Latinize the spelling to add- began in France 15c. but failed there (the Modern French verb is adresser), however it stuck in English. Related: Addressed; addressing.
advertisement (n.)

early 15c., "written statement calling attention to (something), public notice" (of anything, but often of a sale); from Old French avertissement (15c., later respelled pedantically as advertissement, a change rejected in French but accepted in English), from stem of avertir "to turn, direct, make aware" (see advertise). Meaning "public notice (usually paid) in a newspaper or other publication," the main modern sense, emerged 1580s and was fully developed by 18c.; later extended to Web sites.

Advertisements are now so numerous that they are very negligently perused, and it is therefore become necessary to gain attention by magnificence of promises and by eloquence sometimes sublime and sometimes pathetick. Promise, large promise, is the soul of an advertisement. I remember a washball that had a quality truly wonderful—it gave an exquisite edge to the razor! ... The trade of advertising is now so near to perfection that it is not easy to propose any improvement. [Johnson, "The Idler," Jan. 20, 1758]
a- (1)

prefix or inseparable particle, a relic of various Germanic and Latin elements.

In words derived from Old English, it commonly represents Old English an "on, in, into" (see on (prep.)), as in alive, above, asleep, aback, abroad, afoot, ashore, ahead, abed, aside, obsolete arank "in rank and file," etc., forming adjectives and adverbs from nouns, with the notion "in, at; engaged in." In this use it is identical to a (2).

It also can represent Middle English of (prep.) "off, from," as in anew, afresh, akin, abreast. Or it can be a reduced form of the Old English past participle prefix ge-, as in aware.

Or it can be the Old English intensive a-, originally ar- (cognate with German er- and probably implying originally "motion away from"), as in abide, arise, awake, ashamed, marking a verb as momentary, a single event. Such words sometimes were refashioned in early modern English as though the prefix were Latin (accursed, allay, affright are examples).

In words from Romanic languages, often it represents reduced forms of Latin ad "to, toward; for" (see ad-), or ab "from, away, off" (see ab-); both of which by about 7c. had been reduced to a in the ancestor of Old French. In a few cases it represents Latin ex.

[I]t naturally happened that all these a- prefixes were at length confusedly lumped together in idea, and the resultant a- looked upon as vaguely intensive, rhetorical, euphonic, or even archaic, and wholly otiose. [OED]
a cappella 

1868, earlier alla capella (1824), from Italian, "in the style of Church music, in the manner of the chapel," literally "according to the chapel," from cappella "chapel" (see chapel). Originally in reference to older church music (pre-1600) which was written for unaccompanied voices; applied 20c. to unaccompanied vocal music generally. Italian a is from Latin ad "to, toward; for; according to" (see ad-); alla is a la "to the." Sometimes in the Latin form a capella.

Also denoting "that instruments are to play in unison with the voices, or that one part is to be played by a number instruments." ["Chambers's' Encyclopaedia," 1868]

You are not the first person puzzled by the expression "A Capella," or, at any rate, unable to understand it should signify the exact reverse of what it literally does signify. The chorales in oratorios were invariably accompanied, either by double-bass or the whole band. Hence they were, with perfect correctness, said to be performed "a capella." But, as other chorales, sung as part of the church service, were written in the same and simple style the expression "a capella" came in time to be applied to them also, despite their being sung without any instrumental accompaniment whatever. [The Music World, Sept. 11, 1875]