Hospital labs, as the name might suggest, are run by and attached to hospitals. With this setup, the hospital can get tests processed on site, which makes the entire process much faster. Hospital labs also follow the same standards as the rest of the hospital. Doctors can send a wide range of tests sent to the labs for processing.
Outside labs allows hospitals and doctors to focus on direct care while trained lab technicians focus on processing the tests. Private labs are not attached to or run by hospitals. These labs process test results for private-practice doctors, insurance companies, health clinics and clinical-research facilities. These labs may vary in quality, so it is important to look into the lab being used in advance. Check Angie's List for member reviews and ratings of diagnostic labs in your area.
Research laboratories are used primarily for medical research being done by hospitals, universities and pharmaceutical companies, and they also may be used to process rare tests. It is cheaper for labs to send rare tests to labs already equipped to process them than to try to process all possible lab tests.