A discrete channel maps an input symbol $X$ from a finite alphabet to an output symbol $Y$ according to conditional probabilities

$$p(y\mid x).$$

These probabilities form the channel's transition law.

A channel is memoryless when, conditioned on the current input symbol, the current output does not depend on earlier channel inputs or outputs. For a sequence,

$$p(y_1,\ldots,y_n\mid x_1,\ldots,x_n)=\prod_{k=1}^n p(y_k\mid x_k).$$

The binary symmetric channel is a simple example. Each transmitted bit is flipped independently with probability $p$ and received correctly with probability $1-p$.

The input distribution and transition law together determine the joint distribution

$$p(x,y)=p(x)p(y\mid x).$$

A discrete memoryless channel is an information-level model. It abstracts away the waveform, modulation and receiver details that produced these symbol transition probabilities.