They vary. I'm most familiar with burial practixes in the medieval west, where they have changed significantly over the past 2000 years.

In the fourth century, when Christianity was just starting to become wodespread, people in western Europe were often buried clothed, but with few things besides their clothes in their graves. Some people were still buried with coins in their mouths, an old pre-Christian practixe to pay the ferryman to cross the river Styx in the underworld which seems to have continued for a whike after Christianity's rise to preeminance.

By the beginning of the fifth century, it became increasingly common to find people buried with objects connected to their social status within an increasingly militarized society: men were often buried with weapons, and women with jewelry whose style was inspired by military fashions. These burial practices were common in both achristian and non-Christian regions of Europe; indeed, all the evidence we have suggest that burial was still directed bybthe family or local traditions, and not something directed by the church.

The church gradually became more involved in people's funerals, however, perhaps in reaponse to increased fears of purgatory; to avoid suffering after death, many began to invite priests to care for the bodies of the dead, or to bury their dead near saints' shrines or monastic comunities. At the same time, the manner of caring for the body began to change: people began to be buried in simple white garment that imitated the clothing worn by holy monastics. These simple clothes weren't so much a rejection of wearing your 'finest'; they were a different kind of finery, resembling baptismal garments that stressed the holiness and virtue of the deceased. Bodies were also buried with spices, fresh greenery, and other things to keep them as fresh-smelling as possible, with the hope that purifying the dead body could help to purify the soul and save it from purgatory's fires.

The trend to bury the dead in simple white garments - baptism clothes - developed into a simple shrouded burial, in which the dead body was wrapped in a white sheet which again stressed the hoped-for purity of the deceased. It also reflected other social changes not connected with burial practice: the weapons and jewlery which had been placed in graves during the fifth, sixth, and seventh centuries were now being disposed of through different rituals (new inheritance laws, and new taxations systems through which much of this wealth went to the local lord for redistribution rather than remaining with the family of the deceased, or in the grave). These changes had little to do with Christianity (they were social / political), but had the result that all but the most important or wealthy burials became very simple and plain for several centuries.

By the eleventh century, people in some places began to be buried with grave markers which portrayed them as fully clothed and armed, even if their bodies weren't. This practice continued through the high and late middle ages, with many of the more important graves being marked by a statue, brass relief, or carving of the dead body dressed in armor with a sword (if a man), or in a fine gown (if a woman). These effigies were sometimes stylized, but were often very lifelike. Like the burials 1000 years before with weaponry, these graves made statements that were as much about social status as religion: a man's knighthood and status might be displayed alongside symbols of his piety, as a visible reminder to the community of his family's importance, and of the need to continue to pray for his soul.

That's a snapshot of 1000 years of burial in western Europe. Burial during this period was more or less governed by the church (less atbfirst, and more and more as time went on), and there was not a single orthodox burial rite that remained stable over time. Instead, burial practixes changed in response to changing social conditions and changing theology, and bodies moved from being fully clothed to being armed for war to dressed as monks, buried in simple shrouds, and then eventually comemorated with images of clothing instead of the real thing. And, unlike today (where all Americans will probably be buried in a suit or other uniform), social status could matter a lot in how the body would be dressed and commemorated.