This is Shakespeare-era English : “Abraham’s bosom” (KJV) or ‘Abraham’s side’ (ESV) is mentioned in a parable that Jesus told in Luke 16:19-31.
Being “in someone’s bosom” is a picture of being embraced by them (cf. John 13:23, where the disciple whom Jesus loved is ‘leaning on Jesus’ bosom’ [KJV]). The picture Jesus is painting is one where the beggar in the parable is being comforted, even by Abraham, reclining in his embrace, while the rich made faces torments.
The Rich man and Lazarus
The parable Jesus is telling isn’t literal: the weight of the rest of the Bible tells us that when people die they cease to exist and turn to dust (see, ‘Where do we go after we die?‘). The dead do not get taken to Heaven or to a place of horrible torment.
Jesus is simply using a dramatised illustration to make a point. Jesus is saying that people should listen to what the scriptures say — and do what they say — while they are alive. Because it’s too late to do undo this when you are dead.
For more on the the Parable of the Rich Man and Lazarus, see The Rich Man, Lazarus and Abraham, by Steve Cox.