The checksum commands that you have found are used most often for verifying data integrity and tracking information. It's as if you solve a math problem, and then look to the answers at the end of the book to ensure that your solution is correct.
Most common example, is when you download software or Ubuntu.iso image, you will see a file such as this (example from SHA256SUM file for 18.04 release):
When you download
ubuntu-18.04.1-server-arm64.iso file, you want to ensure the file was downloaded OK, that there was no tampering with it by someone in the middle of the network, or that there was a corrupt file downloaded. Thus when you do
you will know right way if the file is OK or not. As for different types, for security applications, the stronger the has
sha512 the better, because attacker cannot break it. Hashes are one-way functions. Input produces hash, but not the other way around. So for security reasons, passwords are best never stored on servers - only hashes.
When attackers steal information from servers, they should only have hash values of the passwords and not passwords themselves. Now, hashes such as MD5 and SHA1 have been broken and attackers can break them to find original passwords. So you don't have to memorize them, but it's good to know if an application is using a strong hash such as SHA256 or SHA512