Yes, a radiologist is a fully qualified doctor. As they are usually "consultants", convention dictates that they are addressed as "Mr/Mrs/Miss" instead of "doctor". Having said that, you will come across radiologists that don't mind being called "doctor" and indeed, some hospital radiologists in the UK and USA tend to refer to themselves as "doctor".
Radiology is the branch of medicine that makes diagnostic images of anatomical structures through the use of electromagnetic ionising radiation or non-ionising sound waves. Radiological imaging techniques include x-rays, CAT scans, PET scans, MRI Scans and ultrasonograms.. Radiology is also concerned with the treatment of disease by using radioactive compounds.
Radiology is used to diagnose a wide range of disorders, including fractures, cancers, ulcers and arterial disease. Sound waves are used to check on the progress of a pregnancy during a "scan". Radiologists are also experts at injecting opaque dyes into hollow organs that allow them to see what's happening in the organs concerned. A radiologist is an expert in the interpretation of these images and will convey his/her opinion to the doctor that requested the imaging for treatment to take place.
As radiologists are doctors, yes they've been to university. They have also spent a number of years after qualifying as a doctor studying radiology in order to become a radiologist - a radiologist without anatomical and disease knowledge gained from studying medicine, would be no use to anyone.