The gittern was strung with gut, had four courses, was played with a quill, and could have a variety of entertaining carvings on the headstock, such as the head of a man, woman or animal, or an acorn (the latter as seen on the Hans Ott gittern, one of only two surviving, which you can see on the link on the main page). The earliest depiction appears to be that painted on a pillar of the Bayeux Cathedral crypt in Normandy, dated to the 11th century. Gitterns appear in the beautifully decorated Cantigas de Santa Maria, a richly-illustrated paean of praise to the Virgin Mary. This song book was compiled in 1260-1280 during the reign of Alfonso X (King of Castilla and other regions in modern day Spain and Portugal). Certainly by the 14th and 15th centuries it was an important and popular European instrument, loved by all classes of people: late mediaeval Italian royal courts recorded the hiring of several gittern masters; Charles V of France's court in the 14th century owned four gitterns; and Geoffrey Chaucer mentions the gittern several times in The Canterbury Tales, written between 1387 and 1400, describing it being played by people who sing and play music in taverns and as a popular instrument to play duets with the lute (which, like the gittern, was played with a quill at this time).
There's not much else anyone can tell you. We have no music for the gittern or for virtually any other specific mediaeval instrument.
Johannes Tinctoris, 15th century composer and music theorist from the Low Countries, stated in his De inventione et usu musicae (Naples, 1481-1483) that the gittern has “the same tuning and method of playing as the lute, though it is much smaller". Previously, the gittern was tuned in fourths, with three or four courses, as was the lute. In the early 15th century, the four course lute had gained a fifth course. By the time of Tinctoris’ writing, the gittern had also gained a fifth course and the lute occasionally now had a sixth. The addition of a fifth course on the lute and the gittern introduced the ‘new’ interval of a third between the second and third courses, what was to become standard renaissance lute tuning.
Iconography shows that the gittern thrived into the last quarter of the 15th century. As Tinctoris testified, the lute was now beginning to be played in a new way, with the fingertips, as well as the long-established quill technique. This reflected a radical change in music-making, from the quill playing of a single line or a melody probably with a drone, to complex and delicate polyphony and counterpoint now possible with the freedom of independently playing fingers. The gittern, it seems, did not make the transition from quill to fingers and was rapidly giving way to the rising popularity of the lute in this new style. The changeover was complete by 1500: there would be no more royally appointed gittern masters, as that role was now to be bestowed upon royally appointed lutenists across the courts of Europe.
Gittern playing a saltarello, anon., Italian, c.1390