Which kind of electrical brain activity is associated with consciousness and why?
Gamma band oscillations (GBO) (Wikipedia) (NCBI)  are 30-90 Hz electrical waves generated by the brain and are thought to possibly be associated with cognition and consciousness (Panagiotaropoulos, 2012). Some evidence for this putative relationship can be seen with experiments such as pre-pulse inhibition (PPI), which can describe how our sensations are interpreted by the brain. Also, PPI may be aberrant in Schizophrenic patients (who have altered cognition). Changes to PPI are associated with aberrant GBO, which could imply an association between GBO and an individual's interpretation of stimuli, their cognition and by extension consciousness.

But of course, the counter-argument is that the intrinsic activity of the brain isn't changed, merely how the brain processes stimuli. However, all forms of measuring cognition that I've studied involve investigating responses to external stimuli, so as far as I know, this is a universal problem. Furthermore, there's a lot we still don't know about consciousness and no model or measurement is perfect. This article gives quite a nice overview on neural correlates of consciousness.