A patent assignment is a transfer or sale of the entire interest in a patent. That means all rights that were originally granted to the patentee transfer to the assignee. Any other transfer of lesser than full rights in a patent is a license.
Assignments can be made before and after a patent application issues as a patent. Within a large corporation setting, applications are typically immediately assigned from the inventor to the corporation so that the corporation can control the prosecution of the application through the Patent Office and so that it can add the eventual patent to its portfolio. However, a patent can just as easily be assigned after it issues as an actual patent.
Patent assignments should generally be recorded to protect against subsequent transfers away from the assignor. An unrecorded assignment is not effective against third party purchasers who do not otherwise have knowledge of the assignment. There is no requirement with the Patent Office that an assignment be recorded, but it is generally a very good idea to do so, especially when the associated filing fees are very inexpensive.