Here's an academic and global perspective. This is deliberately less practical than Josh's answer, but is hopefully a useful reading guide.
I'm also not a lawyer though. You should talk to a proper one to make sure you understand it right, especially because (as we will see) the exact laws differ between countries.
What is copyright?
Having copyright to a thing means being the only one who is allowed to decide when and how it is copied. This means you are allowed to say who is allowed to use it, and how.
This also means you are allowed to give other people some of that right, if you want to. If you ask the copyright holder of a thing if you are allowed to use it for a particular thing or for a particular time, and they say yes, congratulations, you can use it. But you have to ask them, and you have to do it on their terms. It doesn't matter very much how you ask them, as long as you really ask them specifically and their response is clear.
How do I get a copyright?
In most countries, you automatically gain copyright to things that you create, because most countries have signed the TRIPS Agreement, which promises these things.
What happens to my copyright when I die?
The TRIPS agreement says you should have copyright for at least 20 years after you die. Some countries have decided to make that time longer. Here's a list of how long it is in various countries.
Where does a copyright apply?
Your copyright applies in all countries that have signed the Berne convention, as long as it's published in one of those countries first. (That includes almost all countries.)
What copyright exceptions exist?
The TRIPS agreement also requires signing countries to have exceptions to copyright for fair use. The description for this (given by the Berne three-step test) is:
It shall be a matter for legislation in the countries of the Union to permit the reproduction of such works in certain special cases, provided that such reproduction does not conflict with a normal exploitation of the work and does not unreasonably prejudice the legitimate interests of the author.
It's quite vague, so countries have created slightly different variations of "fair use" laws. However, in most places, this includes uses for critique, education and some non-profit cases. Any project that makes money (or makes it even a little bit harder for the copyright owner to make money) very rarely gets an exception.