As a student of mathematics, I learned early on the difference between one, two, and three dimensional objects (or graphs). A one dimensional objects lives in a line, two dimensional objects live in a plane, and a three dimensional object lives in space. This was all intuitive enough. However, the abstractness of mathematics became clear as soon as the various physical dimensions described above began to represent numerical quantities such as color, temperature, market price, etc. Further abstraction becomes apparent when four or more dimensions are added. One may ask, "Why limit the dimensions of a mathematical construct to integers?" That question leads to the development of fractals, which are objects with fractional dimensions. Yet another unintuitive twist is the idea of a zero dimensional object. Such an object would exist within a single point, and would be either on or off.
It may seem that such dimensional analysis has only abstract applications, since as far as we can directly observe, only three dimensional objects seem to really exist in physical form. However, the fact that our world is composed of discreteparticles called "molecules" (or atoms) means that there is more reality to the mathematical analysis of dimensions than our three dimensional intuition may tell us. Granted, much of modern solid state physics deals with three dimensional solids. However, as our ability to analyze and fabricate smaller objects increases, the dimensions of the object being studied may change.
As an example of a two dimensional object, consider a monolayer of surfactant molecules floating on the surface of water, i.e. a Langmuir Blodgett film (LB film). Such a system can be modeled by a two dimensional version of the familiar PV=NRT formula for gases. However, in the case of an LB film the pressure value represented by "P" is a two dimensional analogue to 3-D pressure known as "lateral pressure". As this 2-D lateral pressure is changed, the LB films exhibit several distinct phase transitions between the more familiar solid and liquid phases that we are familiar with in three dimensions. Again analogous to 3-D objects known as liquid crystals, these phases are given names such "nematic" and "smectic," among others.
