In my professor's notes, it is written that if the variable $y$ is explained with an ARMA($p$,$q$) model, then $y_t$ (i.e. $y$ at time t) depends on the most recent $p$ lags of its own value and the most recent $q$ lags of a white noise process.
I understand the fact that $y_t$ depends on the past $p$ values of $y$, but I don't quite understand the $q$ lags of a white noise process. My understanding of a white noise process is that it is a sequence of random variables over time, where each random variable in the sequence is i.i.d. and has a mean of zero.
Can someone explain what it means to depend on $q$ lags of a white noise process? I think it means that at the last $q$ points in time, the value taken by the particular white noise process explains the value of $y$, but I also don't understand the difference between depending on a white noise process and simply having an error term. Is it that the last $q$ values of the white noise process explain the size of the error term for $y_t$?