A copyright owner gets to do, or authorize others to do, the following things: U.S. Copyright Code, 17 U.S.C. § 106
A copyright owner can dole out some or all of those rights to other people or entities, by transferring ownership or granting licenses. Ownership or license rights can be shared by any number of people or entities.
The copyright rights outlined above only apply to works that fit into one of these eight categories: U.S. Copyright Code, 17 U.S.C. § 102(a)
Some of the categories are bigger than you might think. For instance, "literary works" includes almost all text-based media, including computer code.
Works that fit into one of the categories above must also be "fixed in a tangible medium of expression" - that is, saved in some form - in order to qualify for copyright protection. Almost anything counts as "fixed" - a drawing on a chalkboard or whiteboard, or a file saved in a computer's memory would qualify. However, unfixed works such as improvisational speeches or music aren't protected by copyright.
A work also must include original creative expression to qualify for protection. The amount of originality required is pretty low, but just "sweat of the brow" is not enough. For example, writing out an alphabetical list of all Nobel Prize winners may take a lot of work, but it doesn't really contain any original expression. By contrast, if you add annotations and commentary to that list, you could own a copyright in the annotations and commentary.
Copyright does not apply to: U.S. Copyright Code, 17 U.S.C. § 102(b)
More info on what copyright doesn't do, from the U.S. Copyright Office:
Creators today don't have to do anything to get a copyright; a work that qualifies is automatically fully protected by copyright from the moment it is first "fixed". Publication is not a requirement for copyright protection, and even formal registration is purely optional. There is also no requirement to include a copyright notice, date, the "circled c" © symbol, or any other information on the work in order to own a copyright.
Employers usually own the works created by employees in the course of their employment, and commissioned works ("works for hire") are usually owned by the party that commissioned them. Someone who is not a creator may acquire ownership of a copyright through a written agreement with the creator.
Copyrights can be registered with the U.S. Copyright Office by any legal owner, at any time during the term of copyright protection. Early registration (within 3 months of publication, and/or before infringements are discovered) brings some benefits if a lawsuit ever arises. There is a nominal fee for registration, and a copy of the work must be deposited with the Copyright Office. (More info: Copyright Office F.A.Q.)
For works created today, copyright protection starts automatically as soon as a work is created, and lasts for 70 years after the death of the creator.
Because copyright terms have been changed several times, the term lengths for older works vary widely, based on a bewildering number of factors.
Links to resources that provide an overview on copyright laws and policies.