If you assume a zero initial condition (i.e., if the system is switched on at $n=0$, then $y[-1]=0$), then the system is fully described by its impulse response, i.e. its response to the unit impulse $x[n]=\delta[n]$ (which is zero everywhere, except at $n=0$, where it is $1$).
Write the system equation as
In your example $b_0=b_1$. If you apply $x[n]=\delta[n]$, you get the following response:
So we get for the impulse response
The general input-output relation is described by the convolution of the input signal and the impulse response:
E.g., the step response can be easily computed from the impulse response: