Welcome to whitewater boating!
There are certain very popular rivers that require permits from the managing agencies. There is also a misconception outside the boating world that you have to have a permit to run any whitewater or scenic river. The process to obtain permits can range from signing a registry at the put-in to applying for a lottery far in advance, paying fees, and keeping your fingers crossed that you'll be among the 5% of applicants granted a permit.
You can find out about the rivers you hope to run from American Whitewater's Permit Information Webpage
. Another good source of information is the Rivergeek permits webpage
where you can find out about permits required for many Western rivers.
Usually there's information in guidebooks as to whether you need a permit for a particular stretch of river, with info on how to get them.