How does copyright apply to art, especially photography? 

Copyright is the right to control who gets to make copies of your creative work. So, when you, or anyone else, takes a photograph, because it is your creative work, the copyright belongs to you. (This can be tricky when taking pictures of other people, or their art, but the photograph, in its fixed, saved form, is copyrighted to you the moment it is saved. The same thing applies to other photographers and their work.

Copyright is your right to control who uses and publishes your work. If you sell your copyright to a work, the person who buys the copyright can control it; but if you just sell a copy of your work, the buyer does not have the right to distribute work.

Copyright lasts for life of the creator plus 70 years, though older works may go out of copyright sooner, if certain conditions are met. Given the amount of time involved, you can see why photographers and illustrators worry a lot about 'orphan works', where someone wants to use the work but can't find the copyright owner (or, at least, claims they can't.)

How does violating copyright relate to Plagiarism?

Plagiarism is when you take someone else's work, EVEN WITH THEIR PERMISSION, and pass it off as your own. 

Copyright infringement is when you take a work and make a copy or derivative of it without the creator's permission, in a way that doesn't fall under what are called 'fair use' exceptions. (See Below). Using an image in a classroom presentation or paper that you don't distribute over the web is generally considered fair use, for example.