The discrete-time unit step is

$$u[n]=\begin{cases}0,&n<0,\1,&n\ge0.\end{cases}$$

It marks the onset of a sustained discrete-time input.

The discrete-time unit impulse is

$$\delta[n]=\begin{cases}1,&n=0,\0,&n\ne0.\end{cases}$$

A shifted impulse $\delta[n-k]$ is nonzero only at $n=k$. This lets any discrete-time signal be decomposed into weighted shifted impulses:

$$x[n]=\sum_{k=-\infty}^{\infty}x[k],\delta[n-k].$$

Each coefficient $x[k]$ records the value of the signal at one index. The impulse decomposition provides a way to rebuild an arbitrary sequence from elementary signals concentrated at individual indices.