Definition
A stochastic process $(X_n)_{n\ge 0}$ has the Markov property if
$$ \mathbb P(X_{n+1}=j \mid X_0,X_1,\dots,X_n)=\mathbb P(X_{n+1}=j \mid X_n) $$for all states $j$.
Key identity
For a discrete-time Markov chain with transition matrix $P$,
$$ p_{ij}=\mathbb P(X_{n+1}=j \mid X_n=i). $$Each row of $P$ is a probability distribution on the state space $S$, so
$$ p_{ij}\in[0,1], \qquad \sum_{j\in S} p_{ij}=1. $$Consequence
Given the present state, the future is independent of the past. One-step behavior is completely encoded by the transition matrix $P$.