Most users of Power BI will be using it with Power BI Pro licences, but there are other options, and this video explains what Power BI Premium and Power BI Embedded are and when you would want to use them.
Power BI Premium is a composite product that provides a number of different benefits. It changes how licensing works for Power BI: if you buy it, you can move to a model where only content creators need to have a paid Power BI Pro licence, and users who just want to view reports and dashboards can use a Power BI Free licence. It also provides dedicated capacity, which means that it gives you resources dedicated exclusively to your organisation in the Power BI cloud service, and this in turn means there are fewer limitations on performance and data volumes compared to Power BI Pro. Buying different Power BI Premium SKUs gives you different amounts of dedicated capacity in terms of memory and CPU, different licensing options and different features. When you buy Power BI Premium you buy capacities, and in order to benefit from Power BI Premium you have to assign an App Workspace to a capacity. Power BI Premium capacities need to be actively managed by an administrator to ensure that their resources are not over-committed.
Another, separate benefit of Power BI Premium is the ability to install and use Power BI Report Server on-premises. Power BI Report Server is the on-premises equivalent of the Power BI cloud service and is useful for organisations that cannot or do not wish to, for any reason, store their data in the cloud.
Power BI Embedded is a separate Azure service that is intended for use in scenarios where Power BI reports need to be embedded in other applications, for example by Independent Software Vendors (ISV). It is similar in many ways to Power BI Premium but it does not allow users to access reports via a web-based portal, only via other applications, and it allows you to pay according to resource usage rather than by means of a license assigned to individual users.