The disciple/apostle John records in his gospel chapter 19 verses 38 thru 42:
Afterward Joseph of Arimathea, who had been a secret disciple of Jesus
(because he feared the Jewish leaders), asked Pilate for permission to take down Jesus' body.
When Pilate gave permission, Joseph came and took the body away.
With him came Nicodemus, the man who had come to Jesus at night.
He brought about seventy-five pounds of perfumed ointment made from myrrh and aloes.
Following Jewish burial custom, they wrapped Jesus' body with the spices in long sheets of linen cloth. The place of crucifixion was near a garden, where there was a new tomb, never used before.
And so, because it was the day of preparation for the Jewish Passover and since the tomb was close at hand, they laid Jesus there.
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Jesus was not buried. He was covered in bandages with spices and laid in a stone tomb. Later he was not found in the tomb, it was just the bandages (Mary Magdalene discovered) he was then taken to heaven after he talked to his disciples and proved he was Jesus and he lived after death by showing the the piercings in his hands given on his crucifixion.
After Christ's death he was taken down from the cross, which in itself was unusual for the circumstances. Most criminals were left on the cross to become 'food for the crows'. A follower of Jesus named Joseph from Arimathaea was wealthy and influential and used this influence by requesting from Pontis Pilate the body of Jesus to be turned over to him. Pilate relenquished the body, which was wrapped in a buriel cloth, to Joseph who put it in a new tomb that he owned. Jesus died on a Friday afternoon, shortly before the Sabbath. As was Jewish custom to rest on the Sabbath the body did not get prepared for buriel correctly. When the women returned to the tomb on Sunday morning Jesus was gone.
John 19:38-42 - After this, Joseph of Arimathea, being a disciple of Jesus, but secretly, for fear of the Jews, asked Pilate that he might take away the body of Jesus; and Pilate gave him permission. So he came and took the body of Jesus. 39 And Nicodemus, who at first came to Jesus by night, also came, bringing a mixture of myrrh and aloes, about a hundred pounds. 40 Then they took the body of Jesus, and bound it in strips of linen with the spices, as the custom of the Jews is to bury. 41 Now in the place where He was crucified there was a garden, and in the garden a new tomb in which no one had yet been laid. 42 So there they laid Jesus, because of the Jews' Preparation Day, for the tomb was nearby. [NKJV]
This depends on the culture! In the United States, most families pay a funeral home to prepare the body for burial. Many Jewish communities have a burial society, where the members of the society (typically volunteers) do this work. In the Islamic community, members of the immediate family of the same gender as the deceased traditionally wash the body.