“Project Madeira” is a codename for a new cloud offering hosted by Microsoft inside Office 365. The solution is based on Microsoft Dynamics NAV.
You can find information from Microsoft on their blog.
Madeira was orriginaly the codename for the next version of Dynamics NAV as you can read on MSDynamicsWorld.com.
Microsoft has been working on making Dynamics NAV a first class citizen on Azure and in Office 365. With NAV 2015 and 2016 companies can already connect their on premise ERP solution to Azure Active Directory and their Office 365 account and run a hybrid cloud/onprem solution. This works really well for midsize and larger organizations.
With Madeira Microsoft is now taking this one step further hosting the solution themselves and making the onboarding very easy.
For now, Madeira is only available in the US. Which is odd since NAV is mainly succesful in Europe although the US has a strong presence.
Madeira will be offering an extension model allowing customers to add options that makes the solution even more usable for a specific vertical market. Microsoft gave NAV partners the option to experiment with this model with a preview as part of the Dynamics NAV 2016 release.
You can download extensions that help you migrate to Madeira. The standard offering is Quick Books and Microsoft Excel.