It's the state in which an EEG (electroencephalogram) shows that the brain's "alpha waves" are predominating.
It is thought to be a calm, meditative state. Some people try to induce them as a meditation technique.
It is common for scientists discovering new phenomena to use Greek letters to label and identify them; thus, early studies of radioactivity named various kinds of radiation "alpha rays," "beta rays," and "gamma rays." Similarly, the scientists who discovered brain waves named them alpha, beta, gamma, and delta waves.