One measure of the degree of "closeness" of a relationship can be summarized by the number and types of liberties afforded to the parties engaged in the relationship.
For starters, when you meet someone in a higher position than yourself within the same social hierarchy (for instance, student meets teacher), it's common for the student to address the teacher as Mrs. ____ or Dr. _____. After a comfortable relationship has been forged, maybe the student takes the liberty of calling the professor by their first name (or a title, such as "coach").
Another example: it's alright to make fun of personal shortcomings in friends, but not in random strangers. The closer the friend, the more you can say about them without getting your ass kicked. You can also take the liberty of asking progressively larger favors to progressively closer friends. This leads to my "dispose of a body" test of friendship. Ask yourself how many of your friends would unquestioningly help you dispose of a body if you really needed their help. How many of your friends would you help out?
With other types of relationships, there may be physical liberties. For example, I'm a very huggable person...but I won't let someone I don't know give me a hug. That's a liberty reserved for friends/family. I'm sure you can fill in the blank with girlfriends, etc. The closer you are to a significant other, the more you get away with, both physically and verbally (yes, your butt DOES look big).