Password managers are applications that store information for you (conventionally passwords, but they will also work for passphrases, PINs, and security questions). They encrypt that information and use a single strong master password that you enter when you want to access your other passwords. This means that you can use a different, strong password (many password managers will generate these for you) on every site that you log into, without needing to worry about remembering more than one master password. Each of the recommended products can also complete online forms for you, simplifying your entire online experience.
Current security standards do not permit the storing of university passwords in the cloud. Passwords are in the same data classification as credit card numbers, SSNs, Drivers License numbers and health information. For this reason, password managers that have a cloud component such as 1Password and LastPass are not approved solutions for storing university passwords. The only officially approved solution for university passwords is KeePass.