Loosely, it refers to the degrees of freedom an electron has in the material. For example, if you consider the 'density of states' – a quantity which can be thought of as the number of electrons at a given energy level – it has a different characteristic shape depending on whether the material is '1D', '2D', '3D' or even '0D':

http://britneyspears.ac/physics/dos/images/Image441.gif

[As an aside, britneyspears.ac is a surprisingly good resource for a lot of basic solid state physics. You can find a full derivation of the above graph on there.]

Edit: To illustrate, this next graph is the density of states for three different Carbon-based materials: graphite which is '3D', graphene which is '2D' and a nanotube which is '1D'.

http://me3007.wdfiles.com/local--resized-images/thermal-and-... [Source: Hone, J., Quantized Phonon Spectrum of Single-Wall Carbon Nanotubes, Science, 289, 1730, 2000 via http://me3007.wikidot.com/thermal-and-electronic-properties-...]

You can see how these curves are similar to the theoretically-derived curves in the previous graph.

I think there is a different definition of 2D in the materials science field.

http://www.mri.psu.edu/centers/2dlm/

http://www.chemistryviews.org/details/ezine/4976821/Is_Graph...

http://gigaom.com/2013/07/19/graphene-isnt-the-only-2d-mater...

http://www.mrs.org/s13-cfp-o/

My guess is that the people describing it as 2D are focusing on different properties of 2-dimensional space than you are and would agree with you that stacking it would increase the height.

"2D" describes the freedom of the electrons. They can move in X and Y, but not Z. This restricted freedom causes all sorts of unique electronic behavior never seen in 3D systems. You are certainly correct that graphene has height.

Source: I am getting a PhD in 2D electron physics