The answer to this question is both yes and no. You cannot download an iOS or Android app and run it natively on...
your Windows 10 device, but that isn’t the end of the story.
Windows 10 can't run native Google Android Apple iOS or apps, but Microsoft has created two software developer kits (SDKs) that aim to make it easy for application developers to create Windows versions of apps they have written for iOS or Android. One SDK allows iOS developers to port their applications written in Objective C directly to Windows 10. The other SDK is aimed at Android developers and makes it possible to port existing Java and C++ code written for Android platforms to Windows.
Whether app developers will simply be able to recompile their apps for Windows or if they will have to make any changes to the code prior to recompiling the app remains to be seen. In either case, Microsoft's goal seems to be to minimize the amount of work developers have to do to build a Windows 10 app. That will hopefully get app developers interested in Windows 10.
Over the past few years, one of Microsoft's most nagging problems has been a lack of Modern apps for Windows 8.1 and Windows Phone. By giving developers tools to easily port existing apps to Windows, Microsoft stands to solve this problem and take advantage of apps that already exist on other platforms.
Can Windows 10 Mobile improve Windows smartphone adoption?
What is a Universal Windows app?
Compare application virtualization options