The Copyright Act rewards claimants who promptly apply for copyright protection. If a copyright application is filed within three months of publication of the work, then full remedies available under the Act against infringement will extend back to the date of publication rather than begin with the date of filing.
It is not necessary to file a copyright application in order to obtain a copyright. But it is necessary for a U.S. copyright owner, or the owner of a copyrightable work created in the U.S., to register the copyright in order to recover full remedies against an infringer. And it is absoluely necessary for a typical U.S. copyright owner to register the copyright in order to file a lawsuit. If the copyright owner fails to file the application within three months of first publication of the work, then any infringement claim relating to the period prior to filing will not recover full available remedies. Statutory damages and attorney fee recovery are not available for infringement occuring prior to a "late" filing. However, if the copyright application is filed within the three month grace period following publication, then Section 412 of the Copyright Act provides that full remedies are preserved, even as to infringing conduct that occurs prior to filing.
There are some exceptions. Importantly, there is no three month grace period to file a copyright application for an unpublished work. In any event, prompt filing of a copyright application is important to maximize the protection of valuable copyrightable work. The filing fee to the U.S Copyright Office can be as low as $35, so there is no good reason not to file a timely application.