Confused about stationarity and ARIMA processes

The I in ARIMA is for Integrated. The difference between ARMA and ARIMA is specifically that an ARIMA process does not have to initially be stationary (although you have to difference it to be stationary).

First of all random walk is a (0,1,0) ARIMA series. Random walk is a non stationary process. That is why we need this one differentiation. Try it in R plot(cumsum(rnorm(100))) although this plot(diff(cumsum(rnorm(100)))) is a stationary process. I think that you think about white noise as a stationary process (0,0,0).

I am predicting that you thought about Box_Jenkins transformation: https://en.wikipedia.org/wiki/Box%E2%80%93Jenkins_method where is always a step od differencing (,d,) to achieve stationarity. Remember that sometimes one differentiation is not satisfactory to brings it.

Summing up if you have some time series it might be non stationary although by differentiation you could bring it to world of stationary processes. If you want to stay with original time series you could use exponential smoothing theory which not demanding such step. https://en.wikipedia.org/wiki/Exponential_smoothing for example Holt-Winters. However ARIMA covers all space of exponential smooting models.