If you don’t want other people to be able to search for your name and find your Facebook profile using the Facebook people search, you can alter your privacy settings to prevent this. You can also make sure your profile will not appear in the results when people search for your name in search engines like Google and Bing. To adjust these privacy settings, follow the steps below.
Log into your Facebook account
Click on "Gear" image in the far upper right-hand corner of the page
Select "Privacy Settings" from the drop-down list
Click "Privacy" on the left column
Find the catergory "Who can look me up?" where you will find three questions:
Who can look up using the email address or phone number you provided?
Who can look up your timelime?
Do you want other search engines to link to your timeline?
To the right of each question is an "edit" button. For the first two questions above you can choose between, Friends, Friends of Friends, and Everyone. Choose Friends to set the most restricitve settings.
The last question allows to you to prevent your profile from being searchable in search engines. To prevent this, click the edit button which will expand the box. If the box at the buttom next to "Let other search engines link to your timeline" is unchecked, your profile will not be searchable via search engines.
Friends will be able to search for your profile in Facebook, but you can make sure that people you have not added as friends cannot search for you. If you only allow friends to search for you, people who are not your friends already cannot add you as a friend, unless you add them first.