They are basically the same thing, but checksums tend to be smaller (a few bytes).
Both hash functions and checksums are used to verify the integrity of data. Cryptographic hash functions are hash functions for which a collision is unknown. This is why cryptographic hash functions are used to construct things like a MAC (see below).
Another property of hash functions and checksums is that information gets lost during computation. This must be true if you convert some data to a checksum/hash with less bits. This is also why you can't go back to the original data with just a checksum or a hash.
What I think you are looking for is a MAC (Message Authentication Code). Such a code is used to detect the tampering of data. Most of the time it's just a combination of a hash function and some secret value, like a password. See also:
Passwords are sometimes stored as a hash. To verify the password, a hash is calculated of the password you enter, and it is compared to the stored password hash. Checksums are not used for such things because they are generally shorter and more prone to collisions, meaning that you can try random passwords and have a chance that your input has the same checksum as the original password.
But note that using normal (digest) hash functions is not the right way to store passwords. Because they are created for quickly digesting data, attackers can crack those hashes at high speeds. Programmers should use a hash function designed for storing passwords, like bcrypt or Argon2.
Edit: examples of algorithms
To answer your final question about specific algorithms: Please have a look at the Wikipedia page that lists hash functions. Like I mentioned above, they are basically the same. On Wikipedia, checksums are listed as a subset of hash functions.