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My university sent me an email informing me that, during a "periodic check", my password was found to be "easily discoverable and at risk of compromise". As I understand it, there shouldn't be a way for them to periodically check my password unless my password was stored in plaintext. My question:

• Is my understanding wrong, or has my university been storing my password in plaintext?

Your university may not have stored your password in plaintext. They have a very easy way to get the plaintext of your password, and I suspect that they have access to it at least a couple times per day.

You give them your password as plaintext every time that you log on.

They may have used one of your login attempts to check if your password is secure.

However, it's still extremely fishy. Contact your university's IT department and verify that they are storing your password securely. Ask pointed questions on how they checked your password.

And the rest of my advice follows standard internet authentication advice: Do not click on any links in that email; if you do change your password, do so through normal means and not a link that was emailed to you. Use a password manager to store and generate long random passwords. (Ideally, you should only know 2 of your passwords: The one to log into your computer, and the one to log into your password manager.) Never reuse a password for any purpose.

And while you're talking to the university's IT department, ask them about 2-factor authentication.

There are a few assumptions that need to be made here, but what I would imagine that University Password that you refer to, is the password to an Active Directory account. Active Directory passwords deal with passwords in an NTLM hashing format, which are not salted. With this in mind, the same password in different environments will have the same hashed value.

Troy Hunt offers a service called Pwned Passwords that allows administrators to download 517 Million password hashes. It is possible that your school's IT department is comparing the password hashes in their Active Directory, with hashes that appear many times in the aforementioned data.

While storing passwords in plaintext does happen from time to time (mostly in proprietary web applications), the aforementioned scenario would be my assumption as to how they've determined your password is weak.

Your understanding is wrong. If passwords are stored as a strong salted hash, the administrator can’t find good user passwords, but can find ones that are on lists of commonly used passwords by applying the hash and salt to every password on the list and looking for a match. It’s a lot easier if the stored passwords aren’t salted, though, since in that case you only have to run it once and not once per user, so this may indicate that the stored passwords are not salted, which is contrary to best practice.

As I understand it, there shouldn't be a way for them to periodically check my password unless my password was stored in plaintext.

Actually, there is: cracking.

There is a known practice by which system administrators run cracking tools (John the Ripper, Hashcat, etc.) against the hashed passwords. People with simple passwords can be cracked in trivial amounts of time; therefore, as they define it, if they cracked your password, it was easily discoverable and at risk.

To quote this article about John the Ripper:

How you decide to use John is up to you. You may choose to run it on
all the password hashes on your system regularly to get an idea of
what proportion of your users' passwords are insecure. You could then
consider how you could change your password policies to reduce that
proportion (perhaps by increasing the minimum length.) You may prefer
to contact users with weak passwords and ask them to change them. Or
you may decide that the problem warrants some sort of user education
program to help them select more secure passwords that they can
remember without having to write them down.