Well as the other answers and deleted answers have already implied, it’s a bit difficult to say because the question is extremely broad. The answer would change depending on where and when you are (as it would today as well - buying a house can be vastly different even within one country).
First of all, the idea that you can be a person who can afford to buy a house, and that there is a market for houses for individual people or families to live in, really depends on an urban environment and a “middle class”, or at least a class of skilled labour with disposable income. That kind of class and environment didn’t always exist, so we’re limited to places with cities and a middle class - urban areas of England, France, the Low Countries, the Hanseatic towns in the Baltic, northern Italy, for example.
There was a real estate market, sort of, but no real estate industry with agents doing all the work for sellers and buyers. Individuals probably wouldn’t be buying their own houses either - so don’t think of it in terms of modern people buying a single-family detached house. Those didn't really exist unless you were a wealthy lord living in a rural manor. Towns and cities were packed tight with houses, and they were mostly all attached like row houses, with maybe three or four floors if they were big houses. There are lots of surviving timber-framed medieval houses, so that’s the type of thing we’re talking about, more like a small apartment building, with lots of people living in them (also, if you imagine how many of those still exist, imagine how many there used to be! They burned down pretty easily…)
So if individual people didn't buy a whole house for themselves and their families, who owned the houses? Here’s a brief summary for England:
"In larger towns lords owned urban property, ether exploring the houses which had been attached to rural estates since pre-Conquest times, or buying houses as an investment. These were sometimes intended as bases, perhaps for warehouses and accommodation for officials trading on behalf of the lord...Lords also acquired houses which could be rented out profitably to merchants and artisans..." (Christopher Dyer, Making a Living in the Middle Ages: The People of Britain 850-1520, Yale University Press, 2002 pg. 147)
Houses could also be owned by hospitals, churches, abbeys, military orders, etc...i.e. they were usually ultimately owned by a large institution and rented out.
How exactly these houses (or rooms or floors in a house) were rented out or sold could vary enormously, but I can give one example. It’s from the crusader states, so probably similar to how it would have worked in Italy or France. When two people agreed to rent/buy/sell property, including a house or a part of a house, the transaction was usually regulated by the “burgess court”. Burgesses are literally people who live in cities, merchants, artisans, etc., rather than rural lords or peasants (the same word as “bourgeois” essentially). The court approved the transaction and kept a record of it. But it wasn't mandatory, so property could be bought and sold just by a private agreement. If there was any sort of dispute though, it would obviously be preferable to have an official court document. Transactions could also be approved by a church court if the property was owned by the church. Later in the 13th and 14th centuries and beyond, there were notaries to regulate these kinds of transfers, and any other sort of contract. Just like today, you needed a lawyer to do all the paperwork!
Houses could be sold, but so could the ownership of smaller divisions of a house, and any debts associated with it, vifgages, mortgages, etc...this gets into the real gritty details of medieval economics. Mortgages are still familiar today - I don’t know how it works everywhere, but for me, basically a bank gave me a big loan to buy my house and I pay them back in small increments forever and ever until the heat death of the universe. The same idea existed in the Middle Ages.
Otherwise hopefully a medieval economic specialist could comment on mortgages and rent prices...my own specialty in this is really only in the legal sense (who wants to hear about disputes over property boundaries! Anyone? No...OK...)
Here’s some further reading, although I don't know if there’s anything really accessible...this is pretty dry stuff, even for a medieval historian:
Emily Tabuteau, Transfers of Property in Eleventh-Century Norman Law (University of North Carolina Press, 2011)
Marwan Nader, Burgesses and Burgess Law in the Latin Kingdoms of Jerusalem and Cyprus (Ashgate, 2006)
Rodney Hilton, “Some Problems of Urban Real Property in the Middle Ages”, in Socialism, Capitalism and Economic Growth, ed. C. H. Feinstein (Cambridge, 1967)