Each user in an account can import their own sets of data. Depending on how the security tree is set, users can also share data. When user 1 imports their contacts, they will all display Owned By: User 1. That user also has the ability to assign a specific contact, or multiple, to a different user to allow them access to the information.
Assigned To: When contacts are assigned to you, you are able to perform actions on them. You can see their contact details, send them emails, write notes, upload documents, schedule and perform tasks, and even create opportunities. Typically the account owner or an admin will assign leads to their users to follow up with. The contact will always be owned by the original importer, but it can be assigned to anyone to work on.
Owned By: When a contact is first imported it will always be owned by the account that did the import. Whenever a contact is manually created, it will always be owned by the account that did the manual creation.
You can see who owns the account and who the contact is assigned to by opening up the contact details page.