Active Directory Domain Services (AD DS) is Microsoft's directory service in Windows Server. It stores information about network objects such as user and computer accounts, servers and printers, and makes it available to authorised users and administrators. It provides sign-in authentication and access control with a single network account, policy-based administration, a global catalog and replication between domain controllers. Organisations install it on their own domain controllers.
Microsoft is headquartered in Redmond, Washington, USA. AD DS runs on servers you operate, so directory data stays on your own infrastructure in the location you choose, for example a data centre in the EU. Microsoft does not host the data. The vendor remains a US company, which matters mainly for updates and support rather than for where the data is stored.