So the King of England created the courts to resolve disputes in the kingdom. Sovereign immunity was the idea that the king could not be sued in his own courts. We inherited all that. Here, we don't have a king, but the U.S. is a sovereign entity (think of it conceptually as the entity that "owns" the money in the Treasury and also "owes" the national debt). That entity cannot be sued in its own courts.

But even in England, the King would hear certain claims for relief, out of the goodness of his heart. That evolved into the modern idea that the U.S. can consent to be sued, and various statutes of Congress have consented (waived sovereign immunity) for various types of suits. One of those is tort claims (e.g. negligently running over your dog). That's why the DOJ and NSA can be sued for intentional infliction of emotional distress. That's a tort and the FTCA waives immunity with respect to many torts.

The President, etc, are not the U.S. They're just people. They have certain immunities, but they are not protected by sovereign immunity. But that's not enough. For a suit you also need a cause of action (something has to create your right to bring a specific kind of suit). Usually, it's a statute that gives you the right to sue. The case Bivens says that for certain rights that cannot be vindicated any other way, the Constitution itself creates the necessary cause of action.