No, it's a free permit to hike in a park for a day. They want to track trail usage, and sometimes implement quotas to prevent overuse. There are occasionally rangers who patrol the trails and check for permits, but it's pretty rare. You may also need a separate parking permit for national forests (adventure pass), but apparently it's been ruled unconstitutional by the CA supreme court. Many trails in national forests/state parks in California require day hike permits (generally the good forested ones do).

You can see what a dayhike permit application looks like at the very bottom of this page: http://www.sgwa.org/permit.htm. Overnight (camping) permits are separate.