Urbit is a project, not a single computer system. It has three components: Arvo, the operating system; Azimuth, the identity system; and Aegean, the pattern for creating software experiences for individual Urbit communities. This document compares the first two.
Arvo is an operating system that provides the software for a personal server. These personal servers together constitute the peer-to-peer Arvo network. To make this network work on the social level, Arvo is built to work with a system of scarce and immutable identities.
Azimuth is the public-key infrastructure built to be such a system. A suite of smart-contracts on the Ethereum blockchain, Azimuth determines which Ethereum addresses own which Azimuth identities, called points. All point-related operations, such as transfers, are governed at this layer. But Azimuth isn't built strictly for Arvo -- it could be used as a generalized identity system for other projects.
These otherwise-parallel systems meet when you want to connect to the Arvo network. Your Arvo personal server, called your ship, needs to be able to prove cryptographically that it is who it says it is. This proof comes in the form of a keyfile, derived from your point, that you use to start your ship.
A metaphor might help illustrate the relationship between these two systems: the Arvo network is the neighborhood that you live in; Azimuth is the bank vault that stores the deed to your house.