Azimuth is a general-purpose public-key infrastructure (PKI) on the Ethereum blockchain, used as a decentralized ledger for what are known as Urbit identities, or simply identities. Having an identity is necessary to use the Urbit network, which makes it important to have a neutral ledger to determine who owns what.
Azimuth is not, however, part of the Urbit stack. Azimuth is a parallel system that can be used as a generalized identity system for other projects. Azimuth "touches" the Urbit ecosystem when an Urbit identity is used to boot a virtual computer on the Arvo network for the first time. When that happens, the identity considered linked to Azimuth and the identity's full powers are available for use. Once an identity is linked, it cannot be unlinked.
A metaphor might make the relationship between these two systems to understand: Azimuth is the bank vault that stores the deed to your house. The Urbit network is the neighborhood that you live in.