I think they are talking about the ICANN. It's a US organism with some kind of monopoly in the world which control domain names (like
www.city-data.com) and some other stuff. When you write a domain name, the ICANN servers translate this name to a TCP/IP address permitting to know exactly where to find the data of the web page. It's a summary of course.
Most countries would like to end this monopoly because, for example, if this US organism decided to "close" all French websites.. Nobody, in the world, could access them with the mainstream technology (firefox, chrome, etc). Other ways exist like Thor but it's more a niche system currently. So in the end, the US gov could know which website you are visiting wherever you are living. But so far they left the ICANN independant.. They know that if they step in the Europeans and Japanese will freak out. But since the NSA, they don't even need to control the ICANN to know what's going on inside.
About mails it's more because we are all using US mails providers like Hotmail or Gmail. So our datas are stored in the US as well.
Countries like China are worried about that since quite some time, that's why they have their own Youtube, Gmail, Facebook, etc.. But they are still dependent of the ICANN.
The problem I have with the NSA is why ? Why the US gov need all these datas ? Currently it's not a big deal.. But History shows that things change quickly.