A martingale is a sequence of random variables $X_0,X_1,\ldots,$ of bounded expectation such that for every $i\geq 0$,

$E[X_{i+1} \, | \, X_0,\ldots, X_i]=X_i.$

More generally, a sequence of random variables $Z_0,Z_1,\ldots$ is a martingale with respect to the sequence $X_0,X_1,\ldots$ if for all $n\geq 0$ the following conditions hold:

$Z_n$ is a function of $X_0,\ldots,X_n$,

$E[|Z_n|]<\infty$, $E[Z_{n+1} \,|\, X_0,\ldots, X_n]=Z_n$.

See these notes

https://en.wikipedia.org/wiki/Martingale_(probability_theory)

Doob martingale

Azuma-Hoeffding inequality