Conductivity is how well the water conducts electricity. Pure water is not a particularly good conductor; however, when you add dissolved solids these dissociate into ions. Ions allow water to conduct much better. Conductivity is therefore measured to gauge the TDS, or Total Dissolved Solids, of the water.
I don't know of any hobby tests for it, as it is seldom done. We are usually interested in the concentration of a particular ion, rather than the TDS. However, perhaps we should be looking at TDS more? There is a worthwhile debate there methinks.
Conductivity, incidently, is by definition the opposite of resistance. Since resistance is measured in Ohms, conductivity (IIRC from A level physics) is measured in Mhos. Yes, physicists do have a sense of humour.