Never thought I'd get the chance to roll this out anywhere outside of nerdy dinner parties, but my wife is a sommelier and we did a little bit of research together for a wine tasting we hosted at our church. (She covered the wine, I covered the Christianity.)
/u/HelpMeGamer is pretty spot on in that wine then was much less refined than what we get bottled on the shelves today. Wine was pressed manually (or pedally) and was left to ferment in Amphorae, large clay jars that were used for pretty much any kind of liquid transportation in the Ancient Mediterranean. The yeast that effected the fermentation was typically found on the exterior of the grapes that were pressed, and it ends up leaving a much funkier taste and mouthfeel than what we're used to. Combine that with fermenting in damp clay vessels and you'd end up with a taste that most contemporary drinkers would find unpalatable.
While /u/Umbanda is right that yeast will naturally excrete itself to death, in a non temperature controlled environment you'd end up with a slightly higher abv than what we're used to, but that depended largely on the sugar content of the region's grapes.
As far as getting the wine to the table goes it would be ladled from the amphora into a pitcher where it would be cut about 50/50 with water, and then poured tableside into cups. Cutting it with water ensured that you ended up with a more palatable drink that you could consume through most of the day and not get intoxicated.
There are a couple of New Testament anecdotes that shed some light on how wine was consumed in first century Rome. The first one of note is the story of the Miracle at Cana, where John says Jesus turned water into wine. The thing thats illuminating here is that the wedding steward says that folks usually poured the good wine first and then pulled out the bad stuff once everyone was drunk. (This is all in the 2nd chapter of John's Gospel.) This meant that wine was not just wine. There were tiers of quality even in a backwater palestinian province.
In the Acts of the Apostles during the Pentecost event Luke records the crowds in Jerusalem ridiculing the Disciples speaking in tongues saying that "they were drunk on new wine." (Acts 2:13) Meaning wine that hadn't been cut, nor had it been left long enough to mature because they needed to get their fix that badly.
If you're interested in drinking a wine that is a more refined approximation of what the Romans drank you can look for wines with the Italian designation of "Primitivo." This Layer Cake Primitvo is pretty easy to find in stores, but for the real deal (to my mind at least) you want to find one aged in concrete to best approximate a wine aged in an earthen container. If you go to a decently stocked wine shop you should be able to find one no problem.
The yeast bacterias that process sugar into alcohol quit working once the alcohol content reaches 14-18%, so that's as high as ABV can go. You can distill an alcoholic mixture to get non-alcoholic components like water out. This increases alcohol content, as in modern liquors like whiskey or brandy. However, there's little evidence I know of that alcohol distillation was practiced before the late medieval period. So the strength of premodern wine was probably just about the same as most modern wines: 12-15%.
The Roman period was a bit unique, because all of a sudden there were massive urban populations that needed cheap wine (a key source of nutrients and calories: up to a liter a day!), and there were military contracts for cheap wine as well. This led to a mass-production wine industry. As you might expect, the product wasn't always great. Much of it was probably sour or vinegary, like boxed wine that's been sitting out for days (or weeks). The Romans who drank it would often dilute it with water, flavor it with herbs, or sweeten it with honey. No recipes survive, but this is our best guess for the drink they called posca. Wealthier Romans drank swankier stuff. Small batches, local varietals, special processing -- the same quality wines you might expect on the top shelf of your liquor store.
As urbanization declined during the early middle ages, there would have been less demand for the cheap stuff, and the wine industry probably collapsed with the rest of the economy by the year 700. During the depths of this depression, wine was probably kept local, so poor people might actually have had better wine than their plebeian and slave ancestors. Conversely, with the disappearance of a specialized wine economy, aristocrats probably drank less prestigious varietals than their patrician forebears.
After the 700s, the situation began to turn around, and wealthy aristocrats and church institutions were again able to support specialized wine production. Vineyards show up in the early charters of the Carolingian world, and they were definitely a key component of Charlemagne's church reforms around the year 800. All the new churches and monasteries needed vineyards producing wines for daily mass. For large landowners, having a few vineyards was also a way to diversify your estate, so that even if your wheat failed, you still had a shot at a successful season. This situation was ripe for a resurgence of an elite wine culture.
So following the year 800, medieval wine was probably a lot like ours today, with a similar breadth of varieties and values, depending upon the prosperity of where you were at. The situation would change again in the early modern period, when distilling alcohol became popular. By the 1700s, Madeira wine was a booming industry, which required mixing wine and brandy, cooking it, and shaking it up for a long period of time. It included ingredients (distilled alcohol) and processing (constant rocking on global shipping through the tropics) that had not been available in previous times.
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Wine in ancient Rome was far stronger, less refined, and frankly worse tasting than wine today. It was common practice to water down "pure wine" before drinking it. Source: The World of Rome -Peter Jones
Additionally I've heard professors mention that some people would drink wine without watering it down and this was associated with alcoholism but I don't actually have a source for that so take it with a grain of salt.