Jargon is language that is specific to a particular profession or a particular group of people who share a common interest. Oftentimes, though not always, only people from these professions or groups know the meaning of their own jargon. Jargon can be informal or formal, depending on the profession or the group.
Register has to to with the style of one's language: formal as opposed to informal, serious as opposed to easygoing, friendly and familiar as opposed to more respectful and distant.
A register can also reflect the specific style of language one uses. For example, one could speak or write in a business register, a scientific register, a legal-law register, or an academic register.