For the major Medieval battles (meaning, any battle large enough to require this kind of burial, not a border skirmish for example) it was standard to bury the dead in large mass graves in the wake of the battle. There would usually be a few mass pit graves for any given battle. This would be the standard fate of any dead common soldiers.

Nobility had a few options. It's possible they could be transported to a grave in a nearby church depending on where they were from and the region of the battle. For example, in the Hundred Years War it would be reasonably standard for a French noble to buried in a nearby French church. In contrast, if the English wanted to bring a body back to England they had to boil all of the flesh off the body's bones before transporting it. The trip was too long for the body to arrive in England without rotting.

The Battle of Wisby in 1361 has perhaps the most famous mass grave since all of the soldiers were buried with their armour still on. This was because the victorious army's pursuit of their defeated foes lasted so long that by the time they got back to the original battle site the bodies had already begun to rot. Rather than strip the armour off rotting corpses they just dumped it all into pits. This also means this is one of the most important archaeological finds for late medieval arms and armour. The details of this excavation were published in Armour From the Battle of Wisby by Bengt Thordeman

One of the mass graves from the Battle of Towton (1461) was discovered as recently as the early '90s. This grave has only half a dozen or so bodies and so represents only a fragment of the overall dead at the battle. It's believed that these men were killed in the pursuit and dumped into a pit near where they died, some distance from the area where Towton is believed to have been fought. This was all published in the book Blood Red Roses by Veronica Fiorato, Anthea Boylston, and Christopher Knusel.