ArcGIS Server has been at the core of Esri's product offerings for many years. It has provided GIS services to organizations across the world. As described by Esri:
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ArcGIS Server enables the publishing and sharing of GIS resources (e.g., maps, locators, GP models) as web services so they can be accessed and used over the Web/intranet. It is a product that you would typically install and configure on-premise in your organization.
In 2012 ArcGIS Online was launched, a cloud-based GIS "software as a service" or SaaS. Again from Esri:
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"ArcGIS Online" is the online component of ArcGIS. It is a complete, cloud-based, collaborative content management system for working with geographic information. The platform provides an on-demand, secure, open, and configurable infrastructure for creating web maps, web-enabling your data, sharing your map, data, and applications, and managing content and multiple users from your organization.