The short answer is yes, and I want to note that such belief is by no mean restricted to medieval Christians; the earliest mentions of purgatory appear during the first centuries of Christianity, and the doctrine is well-established by the commencement of the medieval period. Note that this post isn't meant as an endorsement of the ideas in the Youtube video, which frankly I don't want to spend the time watching. I'm just answering your questions as presented.

Purgatory was, and remains, an important part of Catholic doctrine. In order to enter Heaven, a person has to be both baptized and forgiven of all sins. Forgivable sins are divided into two categories: mortal and venial. Mortal sins are more serious, and the sinner must confess and gain forgiveness while still alive; purgatory is not involved.

Venial sins, on the other hand, are less serious. Ideally, a sinner would confess and gain forgiveness while alive. However, if a baptized Christian dies with venial sins unforgiven, they will go to Purgatory—their baptism and lack of mortal sin means that they will not receive "eternal punishment" (Hell), but they remain too impure to enter into "eternal reward" (Heaven). Purgatory is most often described as a kind of furnace that purifies the impure material of a sinner's soul, burning away all the sin and leaving only a purified "gold" behind. This doctrine is established by a few key passages in the Bible—the idea of praying "for the dead" and gaining forgiveness for their sins, which appears first in II Maccabees (a text in the Catholic canon, but not in most others).

As for your follow-up: the issue of specific time in Purgatory is tricky during the Middle Ages. The practice of indulgence, through which donations to the Church would effectively function as prayers for the dead (and thus speed up a relative's or friend's time in Purgatory), are tied closely to a popular understanding of Purgatory as something occurring in understandable time; an idea that, to my knowledge, was never an official part of the doctrine. This misunderstanding was only encouraged by indulgence sellers, most famously Johann Tetzel, who advertised that the purchase of a sufficient indulgence might immediately free a soul from Purgatory.

Indulgences are now considered heretical, but the practice was widespread and quite profitable for the Church. Their success is largely owed to the fact that, indeed, most Catholics believed that Purgatory existed, that many people were in it, and that the period of time they'd spend in Purgatory was sufficient enough to justify considerable expense.