What's wrong with this cut on my finger and how do I get it to heal?

The problem with metal slivers is they get stuck and can't back out whereas a wood sliver will still back itself out of a wound like that. The metal has like little barbs on it that don't allow it to back out and just a small piece of it still there will continue to cause your pains. You will probably have to go to a dr. to get that last piece removed.