To designate is to give something a specific status. If you designate your house an opera-free zone, it means that you've officially declared that no opera is allowed to be played there.
To give a person or thing an official status is to designate it as something, like when you designate a meeting place if members of your group get lost at the amusement park. Sometimes, it can carry responsibility or an assignment, like when your teammates designate you as the captain. It can also show a category, like when you designate certain books to the humor section of the bookstore where you work.