you placed 82 posts on blenderartists …and still asking this?
Ok, engine or render-engine is called that part of the program (can also be standalone, just the engine as yafaray or luxrender) that actually makes all the computations that translate the meshes, their materials and the lights into an image. As to say, all the 3d data, and all the parameters converted into 2d pixels. This involves a lot of math stuff…
Cycles is the new engine that blender features since version 2.61
Blender render, often referred as “blender internal” is the old-school engine, coded in the 90’s i guess
as a side note, cycles is expected to totally replace blender internal in the future, when it will get full featured and speed-wise ready
I did not know that, very promising.
Render engine (as game engine) is pure buzzword, i hate them. As already answered, it very simplified and dumbed variant of phrase (a couple of algorithms with all that dirty compromise tricks that used mostly to calculate pixel color from some more high-level data, usually sold as separate product by greedy commercial developers). One better speak to actual algorithms, tricks, their pros and cons, and way to workaround limitations then “OMG that Engine rocks / suxx”.
Blender Internal render is robust, matured, long time developed render using Phong shading model with elements of ray tracing (for shadows, AO), and later extended to use particle systems, smoke. Very integrated with Compositor, you can think it is one “engine”.
Cycles render based in very different model, that cannot be used in past because of more hardware requirements to be practically usable. Time changes, current top hardware can manage that algorithm in acceptable time, and we have more realistic images. Sure, with next generation hardware it dim BI in all places (it still hard to make animation w/o access to real render farm).
Yar(a)ray, Luxrender, Octane, whatever other “plug-in” engines are almost same as Cycles, but developed by other groups, some commercial (Octane), other free, but all need some manual configuration to “plug” to main Blender program, and not integrated to Blender as Cycles for example (no real time in place interaction, visible delays when update data).
If you ask me, Luxrender is more matured, have more features and developer power (and free) compared to all other, but time changes, who know what group make best render ever ?