"Trad" climbing is climbing where as you ascend you place "protection" (little nuts & bolts & spring-loaded cams) into cracks in the rock, and clip the rope to them, so if you fall, you only fall twice the distance from your last piece of protection (assuming your belayer is paying attention and the protection holds).

This in comparison to "sport" climbing, which is climbing on routes where someone has drilled holes and placed bolts on the rock, to which the climber clips the rope. This is more invasive, considered "impure" by many, but is very popular, especially in areas where the rock doesn't have a lot of natural crannies in which to place protection.