The story is called The Double because it centrally involves the protagonist, Golyadkin Sr., meeting his double, a man who looks exactly like him.
This double is a doppelgänger or twin. In Freudian thought, the doppelgänger represents the parts of ourselves we don't want to see. We repress them. Freud insists that whatever is repressed always comes back to haunt us in one way or another. Freud calls this haunting the "uncanny." It is the stuff of gothic mysteries and horror stories, the parts of the world and ourselves we don't want to acknowledge exist.
We know that Golyadkin Sr.'s double is a doppelgänger, because of Golyadkin's reaction to him:
His hair stood on end, and he almost fell down with horror. And, indeed, there was good reason. He recognised his nocturnal visitor. The nocturnal visitor was no other than himself.
We react with horror to a version of ourselves we don't want to see.
Usually, this horrible twin manifests as a Mr. Jekyll, a figure who enacts all our repressed desires to transgress moral codes by killing, robbing, and assaulting, with no concern for other people. We are horrified when this twin emerges, because we have gone to great effort to hide that part of ourselves from our conscious self.
In Golyadkin's case, ironically, the double represents the better self that Golyadkin has repressed: the happy, social, likable self. This is the self Golyadkin's doctor has told him he should cultivate. It is the part that can relax and have a drink, rather than being angry and uptight all the time.
This double appears because Golyadkin's strategy of repressing this better self has not worked for him in terms of career advancement. It represents the extent to which he is unravelling and questioning whether he would be better off letting this repressed part of himself out of hiding.