http://www.youtube.com/watch?v=u1LhGfzJc-AA nice little demo from Martin on moisture measurement basics.
To measure conductivity you generally use a standard cube, dimensions of which are actually not needing specification, just that all 3 must be the same dimension, and the measuring electrodes must be on opposing side of the cube. Thus if you take 2 square electrodes 1cm square and have them 1cm apart you have a standard conductivity cell. Round, square, rectangular ( to certain limits) makes no difference, just the area. Then you measure the desired substance by placing between the electrodes, with the outside being insulated from the material.
Typically for long period measurement you use an AC excitation voltage, as this reduces the effects of electrolysis on the electrodes and the material being monitored. For short term use and single measurements the DC voltage and current from a typical DVM will have little effect, provided you are using an electrode that has an area that will not corrode appreciably during the test.
Conductivity is normally measured for things like water for purity ( normal tap water typically has a conductance over 200 microSiemens, or under 5 kOhm) or for dissolved salt concentration. you can get purified water to incredibly low conductance, lower than 1 nanoSiemens, and this is often used as a high voltage resistor in extra high voltage labs as a current limiting resistor connecting the apparatus to the source. Normally in a 1-2m long glass tube with ultra pure graphite connectors to keep the water from being contaminated. You can touch it while energised provided you are standing on 2m of glass insulator and get the unique feel of all your hair standing up straight away from you, but if you touch ground at the same time ( or are close enough) then you will not have a good day.