Mean p, variance p(1-p). The atom from which the binomial, geometric and negative binomial distributions are built.
A Bernoulli variable is the same object as an indicator variable - the difference is one of emphasis. "Bernoulli" stresses the experiment; "indicator" stresses using it inside a sum to exploit linearity.
Worth noticing: for a Bernoulli variable, X^2 = X, since 0 and 1 are both fixed by squaring. That immediately gives the variance as p - p^2 = p(1-p), which is the quickest derivation.