P(X = k) = C(n,k) p^k (1-p)^(n-k). Mean np, variance np(1-p).
Note the variance is maximised at p = 1/2 and vanishes at p = 0 or 1, which matches intuition - a certain outcome has no spread.
Limits worth knowing. With n large and p small, holding np = lambda, the binomial converges to the Poisson. With n large and p moderate, it converges to the normal by the central limit theorem.
Those two limits are how most interview questions about "many trials" get simplified, so recognising which regime you are in is the actual skill.