Suppose that a group of entities each have public/private key pairs, (PK1, SK1), (PK2, SK2), ... ,(PKn, SKn). Party i can compute a ring signature σ on a message m, on input (m, SKi, PK1, ... , PKn). Anyone can check the validity of a ring signature given σ, m, and the public keys involved, PK1, ... , PKn. If a ring signature is properly computed, it should pass the check. On the other hand, it should be hard for anyone to create a valid ring signature on any message for any group without knowing any of the secret keys for that group.
In the original paper, Rivest, Shamir, and Tauman described ring signatures as a way to leak a secret. For instance, a ring signature could be used to provide an anonymous signature from "a high-ranking White House official", without revealing which official signed the message. Ring signatures are right for this application because the anonymity of a ring signature cannot be revoked, and because the group for a ring signature can be improvised.
Another application, also described in the original paper, is for deniable signatures. A ring signature where the group is the sender and the recipient of a message will only seem to be a signature of the sender to the recipient: anyone else will be unsure whether the recipient or the sender was the actual signer. Thus, such a signature is convincing, but cannot be transferred beyond its intended recipient.