I'm sure what they teach is High German, the German of the media, dictionaries and grammar books.But a person from Kassel might have a noticable Hessian accent. You can tell a New Yorker from a Texan, and likewise you can tell a Hessian from a Swabian even if they both try to speak High German.
For example, in everyday speech I use the structures of High German, but I say "dat" and "wat" instead of "das" and "was" (which you might easily recognize as the English "that" and "what"). Go further north, and people start to say "water" instead of "Wasser" for (you guessed it) English "water", though the first consonant is what you would know as a "v" and not an English "w".
For those who are interested, you probably all know some version of the Lord's Prayer. Here is a collection of the same prayer in various German dialects, which a German from a different place can mostly read and understand and even usually place locally, but you can see that they are very different. The top one is High German, and that's what you'd learn and something which everyone within Germany and Austria should understand. I'm not so sure about Switzerland and Luxemburg (Letzeburg) respectively.
kirchenboerse.de