You've basically got it right that .NET Services is a component that makes up part of the Azure platform. Check out the Azure Services graphic for a visual representation of the components.
Azure is composed of the operating environment and the base services that are used to host cloud applications. It provides the base capabilities (web and worker processes) as well as the simple data storage offerings.
.NET Services are the services that many applications will use to create cloud scale applications, although originally the services were mostly geared toward enterprise development. It includes the access service and service bus.
There are also a few other services that are provided:
SQL Azure is the SQL Server for the cloud offering. It is used like a regular SQL Server instance, for the most part, but is hosted and sericed by Microsoft.
Live Services are consumer focused services that make use of many Microsoft services: Mesh, Live ID, Messenger, etc.
One thing to note is that, although these services are offered as part of the Azure platform, the .NET Services should be usable from any platform. The are REST based services that can be consumed by any application that has the ability to call REST services.