When you receive and accept someone's friend request on Facebook, you will automatically appear in their friends list (and you will gain one friend on your profile as well), and they will receive an email like the one below, confirming that you both are now Facebook buddies:
If you reject someone's friend request though, (and fortunately), the rejection isn't as public or well documented - they will just not receive a confirmation email from Facebook, and you will silently remain "not friends".
Note: when you remove a friend, their friends count will go down by one, you will be removed from their friends list, and they will be deleted from yours.