Trailhead has a nice overview of Page Layouts https://developer.salesforce.com/trailhead/force_com_introduction/customizing_user_interface/ui_page_layouts
Compact Layouts are for the Salesforce Mobile App. Mini Layouts are for Hover links and the Console.
https://help.salesforce.com/HTViewHelpDoc?id=compact_layout_overview.htm&language=en_US
https://help.salesforce.com/HTViewHelpDoc?id=compact_layout_notes.htm&language=en_US
Trailhead has a nice overview https://developer.salesforce.com/trailhead/force_com_introduction/customizing_user_interface/ui_nav_buttons_links
The Stack Exchange questions at Make a button on a layout be visible only to a specific group of users have some great suggestions for buttons and links too.