What does Luke 16:21 mean?

This story of the rich man and Lazarus isn't strictly a parable: it's more direct and less allegorical than a true "parable." But it's not likely to be an account of literal history, either. It's a tale Jesus tells to warn the Pharisees that their greed, unkindness toward others, and intentional disrespect of the Hebrew Scriptures point to the fact that they are going to hell.

An unnamed rich man lives a luxurious lifestyle while Lazarus, a starving, wounded beggar, lies at his gate. Lazarus is apparently immobile as others have laid him there. He is covered with sores and longs for even crumbs, but the rich man ignores him (Luke 16:19–20).

The rich man can't claim ignorance: he knows Lazarus (Luke 16:24). He's presumably aware of a destitute man who's been dumped on his doorstep. Still, the wealthy man does nothing. This is consistent with the Pharisees who would rather break the Law than financially care for their elderly parents. They "dedicate" their money to the temple to exempt its use for their parents, but then keep the money for themselves (Mark 7:9–13).

The name Lazaros means "God helps," as a variation of El'āzār', or Eleazar. In Luke's version of the beatitudes, Jesus promised this: "Blessed are you who are poor, for yours is the kingdom of God. Blessed are you who are hungry now, for you shall be satisfied" (Luke 6:20–21). Jesus also warned the rich: "But woe to you who are rich, for you have received your consolation. Woe to you who are full now, for you shall be hungry" (Luke 6:24–25).

"Desired" is a strong word; Lazarus deeply longs for even scraps. His condition echoes the wise Syrophoenician woman who points out that a family's dog receives the leftovers from the children (Mark 7:28). The rich man doesn't give Lazarus even that. He is so pitiable that he doesn't get the dogs' share, he is the dogs' share, as they lick at his wounds.