Well there are two sides to this question. The beer and who was making it. I'm going to speak very generally as things varied enormously in different times and places.

In Medieval Europe people often made beer in different "runnings." This means that they'd make some beer, removed the wort (unfermented beer) form the spent grain and then they'd add in a little more fresh grain and make the beer all over again. Later runnings were often very very weak. Especially if poor people could get spent grain from a brewery they could make a lot of very weak beer very cheaply. Aside from reusing grain, this beer would often have been especially weak because pre-modern malting (preparing grains for making beer) and mashing (stewing grains in warm water to make wort) were often quite inefficient.

Poor people in the Middle Ages didn't exactly have a lot of cash lying around so they would've often just made the (generally very weak) beer themselves. One of the biggest obstacles to doing that would be that poor people (again, this would vary depending on time and place) often wouldn't have the money to buy a metal brew pot. They could get around this by various techniques such as brewing beer in a big wooden bucket and throwing hot stones into the water to heat it or by mashing in a non-metal container placed in an oven (often overnight).

This beer wouldn't have kept long because of the low alcohol content and because in the early Middle Ages hops weren't used as a preservative. Gruit (an herb mix) was used as a preservative in the low countries but it was relatively expensive so most people just used grain. In any case you have to boil hops for a good long time to maximize their antimicrobial properties which would've been a big problem for people too poor to buy a brew pot. However the beer could end up tasting smoky as many people used wood fires to roast their malt if they couldn't get straw which produces less smoke when its burned. The beer would also sometimes taste a bit of spruce as spruce boughs were sometimes used to filer grain out of the beer.

Pre-hop beer didn't travel well so it wasn't a good commercial commodity except in large cities where there would be high demand. Once the use of hops in beer started to catch on in the later Middle Ages it became a much better commercial commodity since you could put it on a boat and send it to another city with a significantly lower chance of it spoiling. This made beer production much more commercial and hopped beer began to slowly push out older forms of unhopped ale (much to the disappointment of Shakespeare who liked a nice strong ale and hated beer, especially weak beer). This played a part in the rise of commerce in the later middle ages and early modern period before beer became a truly large-scale industry with the rise of porter in 19th century Britain.

As far as who was making the beer you have a lot of different people:

-Peasants making it for their own use: especially early on an in poorer areas this is where a lot of the beer would've come from. People just weren't that plugged into a cash economy and made a lot of stuff themselves.

-People brewing large batches for a special event. This was a good idea as if your beer is going to spoil relatively easily it's smart to brew it fresh for a big event (weddings etc.) and drink it all up then.

-People brewing for their neighbors. If people needed some extra money, were especially good at making beer, or maybe had a nice brew pot in areas and times when those were less common some people would make more beer than they needed and sell it to their neighbors. People would often just come by and drink some beer in their neighbor's kitchen.

-Ale houses. Similar to the above but more specialized and institutionalized (and preferred by tax collectors). These became more common later on in the Middle Ages and how common these were would've again varied a lot depending on time and place. The line between "go to my neighbor's kitchen and drink some of his nice beer" and "go to the ale house and order a few pints" was pretty fuzzy.

-Monasteries were popular with travelers and they brewed a lot of beer locally both for their own use and to serve to travelers. Later on you had secular businesses catering to travelers becoming increasingly common.

-Nobles. The estates of nobles would often make their own beer. Both for their own use (often strong "keeping ale" that was strong enough for the alcohol to hopefully keep bacteria at bay) and to serve up to workers during the harvest as part of their wages.

-Brewers. Full-time brewers would've been mostly an urban phenomenon as that was where the demand was. A lot of the clients of these brewers would be wealthy urban households that didn't brew their own beer. These tended to become bigger and more commercial with the rise of hopped beer. Often where were separate licenses for the (generally bigger) hopped beer breweries and the (generally smaller) unhopped ale breweries and some ale brewers were busted by the tax man for illegally putting hops in the ale without the right license.

Think that about covers it.