Generally speaking, the main recognised categories are:
Free APIs
For obvious reasons free APIs cannot be monetised as part of a monetisation strategy relying on premium APIs. The monetisation of free APIs would rely on one or more of the other strategies.
APIs in which the users are charged
The ‘Consumer Pays’ model can have variants that require the consumer to pay for different levels of usage (Tiered), to pay a fee based on usage (Pay as You Go), to pay according to the consumption of specific units of computing or service (Unit – such as CPU cycles), or to use the API for free but pay for various types of additional services (also popularly known a “Freemium” model).