A Cone rating means that you can fire that clay at any temperature up to that cone. You cannot fire a clay higher than its maximum rated Cone, or it will melt and become deformed. The closer you get to the maximum rated cone, the denser and stronger your clay will be.
Mid-fire stoneware and porcelain, which are the Cone 5-6 clays listed here, can also be used at low-fire or up to Cone 6, but not above Cone 6. Everything mentioned about absorption above applies here too. For dinnerware it is best to use a Cone 5-6 clay if you fire to Cone 5-6.
Remember, you have to fire your glaze to the Cone that is specified for that glaze, regardless what clay you use. Just make sure you use a clay rated at least as high as the glaze.