It's called hedonic regression because it is used to remove 'hedonic' product features (i.e. features that consumers 'like' or get pleasure from), from the price. The hypothesis is that remaining price changes from period to period are caused by inflation.
It is useful where the same product is not sold from period to period - for example the set of houses sold each quarter will be different, with different characteristics (such as size) that have an influence on the price.
Typically for a price index you would fit a linear model with a mix of continous (e.g. size in square meters) and categorical variables (e.g. number of bathrooms) to represent product features, and then use residuals in the index in place of the raw prices. You can also explicitly fit some sort of time-trend to the data as part of the model, especially if you are interested in underlying inflation, and you can use dummy variables to estimate effects of economic shocks, etc.