what volatility do we calculate using GARCH model, Historical vol or Implied vol or Future Vol or Actual vol.
In the case of application in finance, usually, GARCH is used in estimating realized volatility of returns based on the weight we would like to give to each past observation.
Ultimately after estimating (calibrating) the parameters of the model to an existing time-series, GARCH is used for forecasting multi-step ahead return (future) volatility.
Different variants of the GARCH model exist and the application of each one depends on the properties of the time-series to be examined. Some of them can be found Here.
From my understanding I believe you will calculate conditional variance with GARCH. You would then need to take the square root of the variance to calculate the standard deviation/ volatility. One key aspect in GARCH is that you can calculate the "persistence" , I.e. How likely is the asset to "persist" to its long run variance