Naming of quantum chemistry basis sets
Regarding 6-311G**:

The * notation (one for polarization functions on the heavy atoms and ** for polarization functions on the heavy and hydrogen atoms) is an older notation belonging to the split-valence basis sets of John Pople. Many people still use this notation today but if you want to be more precise and more descriptive, the (d,p) notation is what you would want to use.  In fact, we only ever use the explicit (d,p) type notation and never use the *'s in my lab. 

X-YZg

Split-valence basis sets have the aforementioned structure for us to quickly and easily determine how that basis set was constructed. The X simply is the number of primitive Gaussians that describe the core electron shells of an atom. Most of the time you will be freezing core in your computations so you split the core and valence regions of your atoms and assign various basis functions to each one.  The Y and Z components mean you have two sets of primitive Gaussian functions which describe your valence region. Because you have two sets here, you can denote this basis set as a double-zeta. In your case, a 6-311G** is in fact a triple-zeta quality basis set, which is better than a double-zeta. 

For the sake of interest, a 6-31+G* includes a + symbol. This simply means that diffuse functions would be added to the heavy atoms.  A ++ symbol would extend these diffuse functions to the hydrogen atoms. Of course we would never want to use the '*' notation so I should have written this as 6-31+G(d,p).  Of course polarization functions can come in sets as well such as 6-31+G(2df,2pd), and other combinations exist.

Lastly, when using split-valence basis sets such as 6-311G*, you must be very careful to check the manual and see what function space the program is going to use by default.  I will use Gaussian 09 as an example.  By default, Gaussian 09 uses spherical harmonics as its default... EXCEPT FOR A FEW DIFFERENT TYPES OF BASIS SETS including the family of 6-31G* type basis sets which uses the CARTESIAN SPACE BY DEFAULT.  This means that rather than the 5D, 7F stuff that you may be used to using on a regular basis, invoking the 6-31G* type basis set will automatically use the 6D, 10F function set which inevitably changes the size of your basis set which means that you may have data that is not readily comparable to other things.  I always opt out for the 5D, 7F type functions.