Have I Been Pwned is the tool where you search your email, and it displays breaches.
Pwned Passwords is an API (there is a front-end but that's not the usecase) where you send a partial hash of a password to the API, and it returns a list of partial hashes that match, and the implementation from there sees if any of them match the full hash. It's used by quite a few online services to ensure users don't use weak passwords, as if it's shown up in multiple data breaches, they might not let you set it as your password.
https://blog.cloudflare.com/validating-leaked-passwords-with...
As for pwned passwords, it doesn't send the password. The way it works is that you submit the first 5 characters of a hash, and get back a list of all hashes.
You then locally compare the complete hash to see if it is there.
Also, the password doesn't go to the server but its hash goes.
Lastsly, he had to make the free API paid because people were DDOSing it