Normally the more plies you have the more load the tire can carry safely, but you can also make the ride harsher with more plies depending on the type of fabric used in making the plies. On car & truck tires for example the load range letter corresponds to the number of plies used, for example a load range letter D is a 8 ply rating while a F range is 12 ply and the more plies the more load it can carry.

Most bicycle tires today have at least 2 plies, some use Kevlar as the first ply and either cotton or silk (these two are rare today), polyamide, polyester or nylon (more common today) for the second. Most tires I believe now use 3 to 4 plies such as Michelin Axial Carbon, while the Conti Grand Prix 4 Seasons uses 5 plies. But there is more here than just the number of plies, there is also something called TPI (treads per square inch) the more the better, some have only 33 and some have 430. The more expensive tires have a higher TPI count.