I know this is probably a dumb question, but there's a bunch of documentation and I don't know what to do with this stuff. How do you use the API?
Do you mean to say how do I use this stuff?
You can find the documentation here or there's a Get Started section at the right of this page.
For example if you want to see a list of users you could send a web request to http://api.stackoverflow.com/1.0/users and parse then JSON response.
Now if you want to find out details about a particular user lets say your details http://stackoverflow.com/users/566610/jared
Do you see the number 566610 thats the id. Every question, answer, comment ,or user has a id.
So here you need to send your web request to http://api.stackoverflow.com/1.0/users/566610
Your JSON response :
Take one more example. Lets assume we want to find out list of questions for this you should send your web request to http://api.stackoverflow.com/1.0/questions
and if you want questions related to a particular tag for example wordpress
your url will be http://api.stackoverflow.com/1.0/questions?tagged=wordpress
And once you have finished with your application you can register for an API key.
Read How API Keys Work for more details.
You can use StackAPI which is an interactive way to explore the API of the stack family of sites.
Edit 1 :
You can find list of libraries here.
I hope this helps.