P(X = k) = (1-p)^(k-1) p. Mean 1/p, variance (1-p)/p^2.
The mean is the single most reused fact in quant interviews: a fair coin takes 2 flips to the first head, a die takes 6 rolls to the first six.
The one-line derivation is by conditioning: E = p(1) + (1-p)(1 + E), giving E = 1/p.
Memorylessness. Ten tails in a row does not change the expected wait to the next head - it is still 2. The geometric is the only discrete distribution with this property, and its continuous counterpart is the exponential.
Note the large variance: waiting times are far more variable than the mean suggests.