Magnetic resonance imaging uses very sophisticated physics, and is built on many pieces of Nobel Prize winning work.  It is based on the interaction of radiowaves and magnetic fields.  This causes certain atoms – predominantly hydrogen atoms – in tissue to resonate. By emitting different sequences of radiowaves in a strong magnetic field, the scanner is able to detect the pattern of hydrogen atoms (a constituent of water).  It does so as these atoms also emit radiowaves under certain conditions related to the radiowaves emitted by the scanner.  There are then elements within the scanner which detect the radiowaves emitted by hydrogen atoms in the body.
As a result, MRI is able to generate pictures that distinguish clearly between tissues that contain different patterns and densities of water (e.g. between fat and tissue, or between fluid and cartilage).  In effect, the scanner is able to ask a particular point in your body – “what type of tissue are you?”
The rapid development of this technology has enabled modern scanners to ask this question of increasingly small areas giving incredible resolution.  This combination of distinguishing different types of tissue and high resolution gives MRI its advantage over x-ray-based and ultrasound modalities, which cannot either deliver the same resolution (ultrasound) or cannot distinguish as clearly between different types of soft tissue (x-ray based modalities).