Nobody in this thread is coming up with any specific definitions, I don't even think we're all thinking of the same thing.
To me, a breakdown was a change in tempo/riff where, perhaps, you go from some faster style playing to half-time crunching "breakdown".
Example:
Metallica - Blackened...."see your mother put to death, see your mother die"....that whole mid-tempo breakdown was indeed a breakdown.
With regards to groove, in order for groove to exist, the music has to be groovy. Duh. Bolt Thrower, as displayed on their latest album, is PACKED full of groove. That means the drum pattern, in all good conscience, could be thrown into some 70's groovy funk song structure and it would fit. Of course, the key difference being is that in metal, you have a drum pattern groove UNDER the metal riffs. Put 2 and 2 together, and you get groove.