On your home page in Facebook, there's a section for friend suggestions. These are people that, based on various factors, Facebook thinks you might know. Depending on how these suggestions were generated, the other person may or may not see the same suggestion. Note that this is not the same as a friend request -- you still have to friend request the person on your own.
When you sign up for Facebook, you have the option to import contacts from your e-mail accounts. Facebook then suggests as friends all of the people who you imported who have Facebook accounts linked to that e-mail. If that person has you in their contacts list and imported their contacts, he will see the same suggestion.
Facebook suggests some friends based on mutual friends. If you have 50 mutual friends, so the logic goes, you might know each other in real life. When Facebook does this suggestion, it usually is offered to the other person as well. However, they seem to be prioritised by the total number of mutual friends, so if you have only a few mutual friends it may be far down the other person's list of suggestions.
Suggestions From Friends
Facebook friends can suggest friends for you as well. In this case, the other person does not see the suggestion. If you choose to accept the suggestion and add the friend, and the other person adds you, the original friend who made the suggestion will receive a notification.
How to Deal With Suggestions
If you don't know or don't want to add the person suggested, click the "x" to the right of the name. They person will be removed from the suggested list and the next name in the list will bump up to the top. This does not block the person, and you can still add the person later if you want. Accept a suggestion by friend-requesting the person and awaiting his response.