Not a medievalist but I can give you an idea of religious music in that period (can't help with folk music).

Let's start with a warning. We are not 100% sure about the musical practices of that time. At first, this music was not written down. They had an oral tradition, this puts us in kind of a prehistoric situation. The earliest notation comes from around the 9th century, but it is not very specific since it was meant to be useful to remember music you already know. Notation started to become more specific, but we can't still be sure about what they were actually doing.

During the beginning of the period you mention, and earlier, chant was the music of Christianity. There were different traditions coexisting, and then they were kind of competing. Gregorian Chant is the most famous form of chant, there were lots of efforts to standardize it. Our modern system of notation was born out of that process.

Once we have notation, and it starts becoming more elaborate, we start to see polyphonic music. That means music in which you have several voices singing different melodies at the same time (at first the probably weren't soooo different, but the same melody at a consonant interval).

So, that was a basic background, here come the examples. Again, please keep in mind this could sound completely alien to the natives if we visit them with a time machine. Lots of attempts have been made to keep this music "pure," but we are talking about music from quite a few centuries ago.

From the 11th century

Here's some music taken from a twelfth century manuscript that originated in the basilica of the Holy Sepulcher in Jerusalem. Marcel Pérès' ideas have been found a little controversial. He's the guy conducting this and kind of telling the singers what to do. What he proposes sounds different from what most people have done for "medieval music."

Some more music form the 12th century.

Hildegard von Bingen (1098 – 1179) is a peculiar case. You should read about her. We have some music attributed to her, here's about an hour of it.

Moving on, there might have been some guy people called Leoninus (1150s - 1201), composing music at Notre Dame Cathedral.

There was this other guy, Perotinus (c. 1200).

Philippe de Vitry (1291 – 1361).

You've got plainchant which is monophonic, consisting of a single, unaccompanied melodic line. Its rhythm is generally freer than the metered rhythm of later music. Here is a rendition of Dies Irae the word were written in the thirteenth century, thee tradition is much older. It is also normally unaccompanied by other instruments or continuo.

Later on comes Organum where a plainchant melody is augmented with at least one other voice to make the harmony more interesting. For instance you might have a chant melody with another voice singing the exact same thing but a fifth, fourth or octave above or it. Yet the second voice is not truly independent and thus is not what we would call true counterpoint. This was pioneered by members of the Notre Dame school and Perotin especially. Viderunt Omnes thirteenth century.

You've got troubadour music. I honestly don't know much about it, maybe someone will be able tell you more. I simply mention it because it was a thing which youmight want to explore yourself. Volez vous que je vous chant, thirteenth century.

Slowly more instrumental and secular music begins to develop, although Mit Ganczem Willen is technically outside the scope of the question I thought I might as well include it since it lies on the boundary. There were also of course things like fanfares for processions and similar as well.

The earliest music we have for keyboard, probably organ, is from 1360. It contains six pieces, three of them in the form of an estampie (a type of dance), an Italian dance form of the Trecento, as well as three arrangements of motets.