While chemical distiallation as a process was known since Aristotle's time in the Mediterranean world and Middle East, the process was not generallay applied to alcoholic beverages until much later. The earliest (sure) evidence of beverage distillation is in the 10th (North, Jin dynasty) to 12th century (Southern Song dynasty) China and 12 century Italy.
Bench scale distillation of alcoholic beverages might have been practiced for private use by nobility and senior clergy (think monastery abbots), but these beverages would not have been available to other than privileged guests. They would also have been very rough compared to the liquors of today, as aging was not practiced until much later after industrial-scale distillation of beverages began in the 15th Century.
Note that beverages produced strictly by fermentation are naturally limited to about 14% Alcohol By Volume (ABV) as the yeasts die of alcohol poisoning at that point. Wines are readily produced to that strength if the pressed juice contains sufficient sugar, but the final ABV is generally determined from the juice as it is near impossible to stop fermentation once it starts. Brewmasters control the sugar content more directly and readily brew to a variety of strengths.
Beer during the Middle Ages was naturally produced in a wide range of alcohol concentrations, generally classed as strong beers of 8-14% ABV; medium beers of about 4-8% ABV, and weak beers of 1-3% ABV. The latter would have been the everyday drink of the common man and even children into the late 19th century, when water supply quality became reliable.
It is important to note that Medieval beer was nearly always a spiced beer, not a hopped beer, as the flavour and preservation qualities of adding hops to beer was not commonly recognized until the 13th century, and even then spread slowly from Bavaria. (See also What actually happened with beer and Leeuwarden in 1487?) While these various strengths of beer would have been available from the tap in Medieval public houses, one would likely have watered one's own wine to taste after the bottle arrived, in consequence of the different fermentation constraint.
Distillation of alcohol was known since the late antiquity, at least. (Invention sometimes credited to Mary the Jewess, considered by some the inventor of alchemy, 3-d century AD, but Aristoteles already knew the principle). It was practiced in the Eastern Roman empire and then by the Arabs, and penetrated back to Western Europe from the Muslims, as other achievements of ancient science. Originally the product was not a common beverage, but a substance which (al)chemists and later medics used. Gradually it was realized that this medicine not only helps ill people, but to healthy people as well. The earliest beverage was called Aqua Vita, and it spread in the West in 13th century from Italy. There were attempts to ban it, for example, in Muscovy.
It is possible that the distillation process was invented independently in several places. Recently I saw a movie about an African tribe, essentially untouched by civilization which practices this process. It is unclear from the movie whether it was adopted from some more advanced culture or it was an indigenous invention.