Parameters of ARMA model

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 $$y_t$$?

An AR(1) model looks like this: $$y_{t} = \rho_{1} y_{t-1} + \epsilon_{t}$$
Where $$\epsilon_{t}$$ is your innovation/error term/white noise process. Note that $$y_{t}$$ depends on $$\epsilon_{t}$$ but not $$\epsilon_{t-1}$$, $$\epsilon_{t-2}$$, etc. The error term is not correlated across time.
An ARMA(1,1) model looks like this: $$y_{t} = \rho_{1} y_{t-1} + \epsilon_{t} + \theta_{1} \epsilon_{t-1}$$
An ARMA(1,2) model looks like this: $$y_{t} = \rho_{1} y_{t-1} + \epsilon_{t} + \theta_{1} \epsilon_{t-1} + \theta_{2} \epsilon_{t-2}$$
For the ARMA process, $$y_{t}$$ and $$y_{t-1}$$ will be correlated not just because of the $$\rho_{1} y_{t-1}$$ term but because of the $$\theta_{1} \epsilon_{t-1}$$ term. The errors do not disappear immediately, they die out over time. $$y_{t}$$ depends directly on not just one realization of the white noise process, but on the $$q$$ past realizations.