Here’s a quick over view of each concept:
Global illumination is the effect of light bouncing off objects. The bounced light changes the colour of the rendered pixel. How it bounces and how long it bounces is up to the settings you use, but tweak them enough and you’ll get something nice.
Radiosity is the effect of bounced light changing colour slightly depending on the colour of the object it is striking. Say if you have a white plane and a green sphere then there will be a slight green tinge on the white plane where the green sphere’s colour bounces.
Photons are in real life ‘particles’ of light. Mental Ray uses a photon emission system to simulate the passage of light through 3D space and trace the bounced photons to create global illumination, caustic and final gather effects. Other renderers use similar terminology, but I think mental ray is the only one I’ve seen that actually calls it a photon.
HDR is High Dynamic Range. It’s an image that stores dynamic lighting information so that light emission and reflection appears more realistic. You can get away without ever using this, but once you do you’ll find your renders just that bit more realistic.
I’m going to be writing some more tutorials soon, including some on getting your head around mental ray’s bewildering amount of options. You can check out this tutorial I wrote recently if you want to know more about the fundamentals of 3D: