You should consider what your reasons are for conducting a background check. If you just want to find out more about someone, the best place to start is online do use Google or other search engines and do some searches on the name and include any extra information you have to see if you can narrow down your search. The biggest problem with this search is that these searches are very broad and sometimes you don't know if the results are for the person you're searching.
Next you can try more specialized name directories such as whitepages here usually you find information about addresses and phone numbers, sometimes other contact information such as email or social networking links.
Also a good source is social networking sites such as google plus and facebook though again the hardest part is narrowing down the information to the right person but if you do find the right person you can read about friends and post, their interests and style of communication. Also you could find some more details such as where they live or they birth dates.
Next level would be to contact government authorities and public agencies in some states a lot of data is public domain and you can search them in person or online. If you need criminal records then you need to contact the county, police or other authorities to see if criminal checks are available to public.
Though many sites claim you can investigate DMV and driving records usually you can only search and get reports on yourself or your immediate family. Certain data about any individual is private such as medical or driving or even credit. Remember when you apply for a job and they need to get a background check or credit report from you the employer needs to ask your permission. Without a signed permission they cannot investigate properly.
So basically this brings you to realization that if you really need to learn more about someone your only choice is to use an online service. Make sure to use an online service that is reputable and not too costly. All online services offer different kinds of reports such as a full background, or just people search which includes contact information, aliases, address history and sometimes relatives or neighbors. Also when it comes to criminal records you can search one state or nationwide. Obviously if the background is critical to you then you're better off getting a background with nationwide criminal check which checks every state that has information available and only from sources and dates each state makes information available. These are worth the money since they give you a much better and accurate information. Yet keep in mind you can never get a very complete background check unless you are an law enforcement agent and have good reason to run a background check.