Let $Y_1, Y_2, ...$ be independent identically distributed random variables each taking values in $\{0, 1, 2\}$ with the common distribution $(p_0, p_1, p_2)$.
Let $X_0 = 0$ and define
$$X_{n+1} = X_n + Y_{n+1} \pmod 3.$$
Then $X_n, n \geq 0$ is a Markov chain with state space $\{0, 1, 2\}$ and transition matrix
$$P = \begin{bmatrix} p_0 & p_1 & p_2 \\ p_2 & p_0 & p_1 \\ p_1 & p_2 & p_0 \\ \end{bmatrix} $$
Questions: What is meant by the common distribution? Why is $P$ the transition matrix?
It appears that the $P(0, 0) = p_0; P(0, 1) = p_1;$ and $P(0, 2) = p_2$. Is this obvious by the common distribution? And why are the other rows calculated in that way?
$Y_i$ represents the increment, label $3$ vertices with labels $0$, $1,2$ in clockwise direction. $p_0$ is the probability of being at the same spot, $p_1$ is the probability of moving $1$ step clockwise and $p_2$ is the probability of moving $2$ steps clockwise.
Incrementing by $2$ is equivalent to moving clockwise by $2$ steps, which is also equivalent to moving anti-clockwise by $1$ step. Note that $2 \equiv -1 \pmod{3}$.
Since $p_0$ is the probability of being at the same spot, hence the diagonal entries are $p_0$, and the position to the right (modulo $3$) is $p_1$ and the position to the left of the diagonal (modulo $3$) is $p_2$.