The backshift operator $B$ is used to rewrite auregressive models as factorable polynomials. The backshift operator shifts a time series back one time step.
$$ B X_t = X_{t-1} $$When the backshift operator shifts more or less than one time step, this is indicated with a exponent. $B^n$ represents the n$th$ backshift. Thus, more generally
$$ B^k X_t = X_{t-k} $$For a single step back in time $B^1$, the exponent it typically omitted.
The following are real application of using backshift notation.
The zero-meam AR(1) is expressed as
$$ X_t - \phi X_{t-1} = a_t $$This can be rewritten in backshift notation as
$$ (1 - \phi B) X_{t} = a_t $$The first order difference filter is expressed as
$$ X_t = Z_t - Z_{t-1} $$This can be rewritten in backshift notation as
$$ X_t = Z_t (1 - B) $$The 3 point moving average filter can be expressed as
$$ X_t = \frac{Z_{t+1} + Z_t + Z_{t-1}}{3} $$This can be rewritten in backshift notation as
$$ X_t = Z_t \frac{B^{-1} + B^0 + B^1}{3} $$