The Medieval period in Western classical music lasted from around 486 AD (the fall of the Roman Empire) to around 1400 (the beginning of the Renaissance).
During the early Medieval period there was no system of notation for writing down music. The predominant music of the time was Gregorian chant, in which monks chanted in unison. These chants began to be written down with an indication of the basic pitch levels (i.e. whether the notes were moving higher or lower), but with no indication of rhythm. Over time, harmonies and accompanying lines were added, which in turn led to polyphonic music (multiple voices singing different things at the same time) such as the motets that were popular around the 1300's. As the music became more complex, the written notation became more precise, with a basic two-line or four-line staff to show the intervals between notes. In addition, the notation began to indicate a basic rhythm (how many beats for each note). Written music had to be hand-copied by scribes, which was expensive, so most of the written music from this period comes from churches and monasteries.
By the late Medieval period, it became popular to use complex musical styles in which the spirit behind the words was more important than the words themselves. One way this was done was by singing multiple notes for a single vowel, which made the words difficult to recognize because vowels were being stretched out over several seconds (this is referred to as "melismatic" singing). Another method was to stick to a rigid rhythm which was continuously repeated, making the individual words difficult to recognize (this is referred to as "isorhythmic" singing).
The Medieval period also saw the rise of various types of troubadours, who were skilled musicians, poets, and singers. They typically sang songs about war, chivalry, and courtly love. Courtly love was a code of behavior in which love was the highest virtue, and the lady in the relationship held the power to decide when she and her love would meet or what they would do. Her love was considered of little value if it was easy to win, and her love was highly valued if it was difficult to win.
These are some of the composers of this period, and a taste of their music. The links will take you to the Wikipedia pages for the composers and their works.