I'm getting through but getting a message "Please note: due to exceptional demand the site may run a little slowly today. Please check back soon!". The site has in fact been up and running for at least a year now (I already had it on my list of links on the main group page!) but perhaps there was a lot of work to be done behind the scenes before the official launch.
From what I understand, TNA = The national Archives, E=Exchequer Records, and then the rest are subcategories within the Exchequer records, down to m6= membrane 6.
The National Archives site puts forward slashes in where the underscores are in that listing, but I'm guessing when they designed the database, forward slashes were not allowed for computery reasons (I think that's the technical term :)).
There is a search facility where you just enter in the catalogue number, which would likely be E 101/44/6. Though I remember having problems with whether there were spaces or not when I searched there. You can also browse from the E heading down through each category until you reach your section.