Nokia X phones run on the Android Open Source Project (AOSP) version of Android. Specifically, the Nokia X platform includes the Linux kernel, the open source pieces of Android (Dalvik, general open-source Android libraries and the application framework), extensions that Nokia has made to that framework, and Nokia application programming interfaces (APIs) that Nokia built to replace Google Play store APIs.
Nokia built APIs for maps, push notifications and in-app billing that replace Google's comparable APIs. And Microsoft (and its Mobile/Nokia arm) developed versions of some of its key apps and services, including Skype, Outlook.com, OneDrive and Nokia Camera. Nokia X users can download more Android apps from the Nokia Store.
A related aside: I've seem some speculating that Microsoft is planning some kind of a Windows Phone launch on or around June 24, which, presumably, would be for Windows Phone 8.1. This is not happening, according to my sources.
Windows Phone 8.1 Update 1 is rumored to be coming later this summer/early fall. It is expected to include new 3D gesturing technology that will allow users to hover over tiles and apps to interact with them. I believe users will need new hardware in order to make use of this capability, however.