The format varies. Many firms just charge a straight fee as a percentage of assets, and/or as a percentage of in/outflows e.g., mutual funds.
For fund managers that are not hedge funds, returns are compared to index returns if there are any incentive bonuses. Individual fund manager’s bonuses are often based on this relative performance, even if the client is not charged. Better returns presumably brings in more clients for the fund, and so the fund firm is more profitable, even if it does not charge performance fees.
The hedge fund 2/20 model was 2% of net asset value, plus 20% of returns in excess of some target (discussed next). There would be other mechanisms, such as a “high water mark” rule, which would means that the variable compensation would not be paid until any losses are recovered. The 2% fixed fee always needed to be charged to keep the expensive hedge fund operation going. Since hedge funds typically used leverage, the 2% would have been smaller as a percentage of assets.
The exact terms would be negotiated, the stories that I have seen is that typical terms are now less generous than the 2/20 numbers, but that varies from fund to fund. Very popular hedge funds might charge more.
Hedge funds were supposed to diversify from equity market risk, so the return target was often either fixed (e.g., 8%), or a money market rate plus a spread (e.g. Treasury bill returns plus 5%). That would change based on the structure of the fund.
As an example, if the target rate of return is 8%, and the fund returned 13% (5% more), the fund would get 20% of the 5%, which is 1%.