The conductivity of a solution is simply a measurement of how much ions (charged particles from dissolving salt, acid or base, etc) are in the liquid. The more ions there are, the more conductive the solution is. This is basically because more ions means there are more particles to carry electrons, the most fundamental components of electricity, around, making the solution more conductive to current when a voltage is applied.
A conductivity probe consists of two main parts: the anode (positive for a power consuming device) and the cathode (negative) which are usually made from an inert material such as graphite and part of a single contraption. This is then placed into the solution in question with the other end connected to a measurement device such as a computer. The probe is activated when a voltage is applied to the system. This puts forces onto the charged ions in the solution which push them to either the anode or cathode depending on their charge (negative, electron carrying ions moving towards the anode and positive, electron-less ions moving towards the cathode). As the negative ions reach the positive anode, their electrons are taken from them and they become positive and start moving towards the negative cathode. At the cathode they are given an electron and the process repeats.
What the computer measures is the conductivity of the solution by measuring the amount of current (proportional to ion concentration) when a potential difference is applied between the two parts of the probe. We can then tell when different solutions have different conductivities when they give different levels current given the same voltage. This is then converted to conductivity by multiplying by the distance between the anode and cathode divided by their total surface area submerged in solution.
This conversion is necessary because as the distance between the anode and cathode increase, it becomes harder to pass electricity through which decreases the amount of current without doing anything to the solution. Thus, we need to multiply by the distance to compensate. On the other hand, if we make the two electrodes larger, it becomes easier to pass a larger current through. Thus, we need to divide by the surface area to compensate.