The "extra dimensions" are simply extra dimensions of space, in addition to the usual three (x,y,z). There are a few differences, though. First, they are not "straight" dimensions, but are curled up into tiny circles. Second, only gravity can enter these extra dimensions. Light and other particles can't, which is why we can't readily see them.
Don't lose your enthusiasm! IF the LHC finds evidence that extra dimensions really exist, that would be the most exciting result possible to come out of the experiment! The other expected results -- finding the Higgs boson, or possibly supersymmetry -- will merely show us that new particles exist. But finding extra dimensions would radically alter our concept of the universe. You can't help but be excited about that!