I assume you are referring to sites that allow one to do a background check on another person. There are many sites that provide free lookups for telephone numbers and street addresses. Internet search engines can be used to mine for information about a person that has appeared in a newspaper, newsletter, blog or other online posting. And there are many online services that provide fee-based access to public records, such as property ownership, voter registration and court filings. Online Sex Offender Databases can tell you if there is a sexual predator in your neighborhood. There's no way to find out when someone is digging for these types of public information about you, because the law permits anyone to access it.
For the non-public information, such as criminal and financial records, there are federal and state regulations that limit the types of information that can be disclosed, and to whom it may be released. Employers in some fields (especially those that deal with children) are required to run comprehensive background checks that report on a person's criminal record, credit standing, character, general reputation, personal characteristics, or mode of living.
It's also common for landlords, lenders and insurance companies to run a credit check when they are considering entering into a business relationship with a given person. Credit files maintained by consumer reporting companies include information about you such as mortgage, credit card and loan balances, along with your payment history.
Employers must obtain consent from the applicant before running comprehensive background checks, and on your credit report you can see a record of everyone who has received a copy of your credit file within a certain period of time. See my article Free Credit Reports to learn more about credit reports, and how to get a copy of your credit file.