Magnetic Resonance Imaging is an umbrella term for any medical imaging technique that makes use of the phenomenon of nuclear magnetic resonance. There are many variants depending on how the scanner is configured and analysed, but all are considered MRI.
The basis for the different types of MRI scanning
Much of the following is adapted from MRI Basics.
Essentially, the protons in hydrogen atoms in the water that constitutes your body are sensitive to magnetic fields and align when placed in a very powerful magnetic field. Short bursts of radio waves knock them out of alignment. When the protons realign, they emit radio waves again, which is what is detected by the scanner. This frequency data is processed using a Fourier transform to get the relative intensity at each point in the plane.
Different variations are possible by altering the frequency and timing of the radio energy pulses applied. This gives various sequences such as T1-weighted, T2-weighted, Fluid Attenuated Inversion Recovery (Flair) and diffusion weighted imaging (DWI), which differentiates tissues based on the movement of water. Each is better at evaluating a particular type of body tissue, distinguishing between certain adjacent tissues or detecting a disease process such as inflammation.
This image shows how the brain would appear under T1, T2 and FLAIR sequences.
Gadolinium can be used as a contrast medium, particularly to identify vascular structures. Unlike the radiopaque contrast needed for certain types of plain x-ray or CT, gadolinium (when chelated) is paramagnetic.
Types of scanning procedures
There are many types of MRI scan, depending on the anatomical area under investigation and the configuration of the scanner, as detailed above. Here are some examples:
Magnetic resonance angiography (MRA) - using MRI to identify the course and potency of blood vessels.
Magnetic resonance cholangiopancreatography (MRCP) - used to investigate the biliary tree and pancreas (less invasive than the endoscopic alternative, an ERCP).
Cardiac MRI - imaging the heart requires pulsed sequences synchronised the the patient's ECG.
Functional MRI (fMRI) - this depicts changes in deoxyhemoglobin concentration due to modulation of neural metabolism. It is used to identify metabolically active areas of the brain, as opposed to investigating anatomical pathology. It has been used for studies in neuroscience, surgical planning, for monitoring treatment outcomes.
Magnetic resonance neurography (MRN) - as mentioned in the question, this is an adapted form that is especially good at distinguishing nerves from surrounding tissue by optimising the configuration for the water properties of nerves. This was initially an adapted form of diffusion-weighted imaging that is configured to focus on the fact that water tends to diffuse anisotropically in nerves (it diffuses more prominently in the longitudinal axis, along the length of the nerve). It has been found to be a useful adjunct to other means of assessing peripheral nerves such as nerve conjunction studies and electromyography. See this study.
Here is an example image of a magnetic resonance neurography study. It demonstrates the brachial plexus.
It is likely that an MRN will be done if a nerve needs to be directly assessed (e.g. checking for a specific nerve injury like a nerve root compression of the spine (radiculopathy) or carpal tunnel syndrome). MRN would not automatically be performed when imaging a body area for other reasons (e.g. MRI of chest for lung disease etc). It should be clear on the radiologist's report which sequences and configuration were used.
In summary, the term MRI covers all types of nuclear magnetic resonance imaging. MRN is one example.