The copyright of a work of art automatically belongs to the creator. Even if the physical piece of art is sold to another person, copyright does not transfer. This means the new owner can only display the work of art, but cannot take pictures of it and reuse them online, in a print work, or in any other form. If the rights holder allows a digital image of their art to be viewable online, this does not mean the viewer can download and use the digital file.
This is a complicated subject, but a simplified answer (for the US) is:
Works in the public domain are free for anyone to use or reuse in whatever way they'd like, with no requirement to cite. Anyone can use a public domain work without obtaining permission, but no one can ever own it. Older works are in the public domain because their copyright has expired. Copyright holders can also purposely put their work in the public domain to allow others to use it freely.
Fair use permits limited use of copyrighted works without having to acquire permission from the rights holder. There are no firm answers about what constitutes fair use; rather, it is decided on a case-by-case basis by evaluating the following four factors:
Please watch the following short video from the Library of Congress that explains Fair Use.
If you would like more info on copyright, including its history and purpose, please view the following video, also from the Library of Congress.