A domain represents an entire customer - e.g. an Institution, Consortium or Company - who subscribes to the OpenAthens service. You will have one administrator account that represents the domain. This login works the same as any other administrator account but has one additional preference pane for things that cover the whole domain.
A domain may or may not have any organisations beneath it.
An organisation represents part of a domain that has its own permission sets and user accounts. These are sometimes called sub-organisations or sub-administrators.
An organisation might exist where a domain has regional locations, e.g. a multinational company might have offices around the world, a health service might comprise several hospitals, or a University might be spread over different campuses.
An organisation does not subscribe to OpenAthens directly - the subscription is managed by the domain administrator.
An organisation will sometimes have unique identifiers for subscriptions to enable different parts of the domain to subscribe to content separately, but will usually inherit the identifiers of the domain.