Domains
The format of the base URL for Modrinth's API is https://{branch-}?api.modrinth.com/{version}/, where branch is the branch of Modrinth you're using and version is the version of the API you are using (the latest version of which is v2). You can learn more about how Modrinth's API versions work here.
Version 1 of the API is the only exception to this rule, as it was the following: https://api.modrinth.com/api/v1/
Version 1 is deprecated, and anything using it should be migrated to version 2. See its migration guide here.
The branch can either be staging or nothing (production). If you're using the production server, the path will be https://api.modrinth.com/.... These use separate databases, with staging being primarily for testing the API before it is rolled out onto the production servers. Production uses the latest tagged on the labrinth repository, whereas staging uses its latest commit.