Inqubu has recently published a public data explorer called INQStats. With INQStats, statistical data from all countries of the world can be explored and compared. Now, a public data API was published as well which can by used by developers to get up-to-date demographic data for different countries of the world. The API is based on REST and can be accessed by the URL http://inqstatsapi.inqubu.com/. Compared to other demographic data services such as Google Public Data Explorer, the amount of available data sources is more limited, but the user interface is more advanced and data from INQStats can easily be added to other apps or services by the interface (return format is JSON).
If you are a developer and need statistical or demographic data like median age, median income, Gross Domestic Product, etc. for all countries of the world, you should read this blog post where the guys from inqubu explain in detail how to access their API.