The invisible area around any person which is off-limits to others.

Generally considered to be a foot or two here in the U.S.
There are, of course, special occasions. For instance when standing in line people can be closer, perhaps even touching, and not be considered awkward or uncomfortable.

Usually used in reference to someone's invasion of a personal bubble. Often when someone who is smelly, disgusting or disliked is trying to make friends, and not succeeding.