Shibboleth is federated identity management software. With federated identity management, institutions join together in a group—a federation—and agree to trust each other's identity credentials for logging in to websites. Shibboleth is an Identity & Access Management service.
Shibboleth allows people to log in to web resources at other institutions using the ID and password they use at their own institution. It's kind of like when banks allow you to use your ATM card at the ATM of a bank where you don't have an account.