Laudator Temporis Acti

As a rule, the ancient Greeks and Romans did not drink their wine pure, but mixed with water. The Romans called unmixed wine merum, from the adjective merus (unadulterated, whence English mere). The Greek equivalent is the adjective akratos (unmixed), from alpha privative and the verb kerannymi (mix). The Greek word for a mixing bowl, krater (whence Latin cratera and English crater), comes from the same verb. In English we use the word neat, meaning without admixture or dilution.

It was considered barbaric to drink wine neat in ancient times. Herodotus 6.84 (tr. George Rawlinson) tells this cautionary tale:

The Argives say that Cleomenes lost his senses, and died so miserably, on account of these doings. But his own countrymen declare that his madness proceeded not from any supernatural cause whatever, but only from the habit of drinking wine unmixed with water, which he learnt of the Scyths. These nomads, from the time that Darius made his inroad into their country, had always had a wish for revenge. They therefore sent ambassadors to Sparta to conclude a league, proposing to endeavour themselves to enter Media by the Phasis, while the Spartans should march inland from Ephesus, and then the two armies should join together in one. When the Scyths came to Sparta on this errand Cleomenes was with them continually; and growing somewhat too familiar, learnt of them to drink his wine without water, a practice which is thought by the Spartans to have caused his madness. From this distance of time the Spartans, according to their own account, have been accustomed, when they want to drink purer wine than common, to give the order to fill "Scythian fashion."

Likewise Plato, Laws 637 E (tr. Benjamin Jowett), says:

But the Scythians and Thracians, both men and women, drink unmixed wine, which they pour on their garments, and this they think a happy and glorious institution.

An exchange between one of the ambassadors (A) and Dicaeopolis (D) in Aristophanes' Acharnians (73-78, tr. Alan H. Sommerstein) also illustrates the barbaric nature of the custom:

A: And when we were entertained, we were compelled to drink unmixed sweet wine from cups of glass and gold -- D: City of Cranaus! are you aware how these ambassadors mock you? A: Because the barbarians regard as real men only those who can eat and drink vast quantities.

A Greek or Roman who drank wine neat was more likely than not a drunkard or a glutton. Martial 1.11 (tr. Walter C.A. Ker) criticizes one such individual:

While twice five wine-tokens are a knight's allowance, why do you, Sextilianus, all to yourself take twice ten drinks? By this time the warm water would have failed the attendants who bring it, were it not, Sextilianus, that you drink your wine unmixed.

Cum data sint equiti bis quina nomismata, quare
  bis decies solus, Sextiliane, bibis?
Iam defecisset portantis calda ministros,
  si non potares, Sextiliane, merum.

The 21st epigram of Ausonius is a labored pun on an old woman named Meroe after the Egyptian city of that name. After giving several examples of "significant names," Ausonius ends the poem by claiming that Meroe's name is really derived from merum (unmixed wine):

And so you too, Meroe, not because you are black in color, as one who is born in Meroe on the Nile River, but because you do not dilute the wine poured into your cup with water, accustomed as you are to drink unmixed wine, pure merum.

et tu sic, Meroe, non quod sis atra colore,
  ut quae Niliaca nascitur in Meroe,
infusum sed quod vinum non diluis undis,
  potare immixtum sueta merumque merum.

According to Xenophanes (fragment B 5 West, tr. J.M. Edmonds) you're supposed to put the water in the mixing bowl first, then the wine:

Nor would a man pour wine first into the cup when he mingled it, but water and thereafter the liquor.

It was the function of the master of the drinking (Greek symposiarchos, Latin magister bibendi) to decide the proportion of water to wine. The master of the drinking was elected by his fellows (Xenophon, Anabasis 6.1.30) or chosen by lot (Horace, Odes 1.4.18 and 2.7.25-26).

Three parts water to one part wine is the proportion recommended by Hesiod, Works and Days 596, although the proper proportions were a matter of much dispute in antiquity. Aristophanes, Wealth 1132 (tr. Alan H. Sommerstein) mentions one part water to one part wine, a strong mixture:

Ah me, the cup of fifty-fifty blend!

Athenaeus in his Deipnosophistae (Professors at Dinner) goes on for pages (10.426b-427a, 10.430d-431b) quoting various authorities on the question. Most recommend more water than wine, although Alcaeus, fragment Z 22 (tr. Denys Page) favors more wine than water:

The son of Semele and Zeus [Bacchus] gave wine to men for oblivion of sorrow; mix one of water to two of wine, pour them full from the brim down, let one cup jostle another.

See Denys Page, Sappho and Alcaeus (Oxford: Clarendon Press, 1955), p. 308, for a learned discussion of proportions of water to wine in antiquity.

A variation on this ancient custom is performed even today as part of the Mass, when the priest pours a few drops of water into the wine and says this prayer (Novus Ordo):

By the mystery of this water and wine may we come to share in the divinity of Christ, who humbled himself to share in our humanity.

Per huius aquae et vini mysterium eius efficiamur divinitatis consortes, qui humanitatis nostrae fieri dignatus est particeps.

That this has been part of the Mass since early times is shown by the evidence of Justin Martyr in his first Apology (chapter 45), written around 150 A.D.