For the 2004 and newer, the VW 505.01 (or newer) spec is rather important (although for the 2004-2006 TDIs many people have found that the Mobil "Turbo-Diesel Truck" works superbly)... for the 2009 and newer TDIs, the VW 507 spec (or an equivalent Mercedes rating) - chiefly related to the low ash content required by the new emission equipment - is mandatory.
For the 2003-and-older ones, the minimum spec is VW 505.00 (which is a manufacturer-specific rating) which is related to the European ACEA A3 spec (see your owner's manual)... Also listed is a North American API minimum of (I believe) CG-4... Most oils will only have the API or the European standards... not both... Since the CG-4 rating is quite old, the newest oils, CH-4, CI-4, CJ-4 do an even better job, and since the Rotella and Mobil are readily available, they're usually recommended. (BTW: Petro-Canada "Duron", Chevron Delo 5W-40, and one or two others are also highly regarded...)
Hope that made some sense... (I haven't yet had my 2nd cup of coffee this morning...)
Yuri
PS: You asked whether you should use the 507 oil... I believe that the 507.00 oils are 5W-30, and although the very modern oils are more flexible, many consider a 5W-40 oil to be the better choice, as it protects the top rings and turbo better...