The definition comes from the way the cords (the string like pieces that lie under the tread that give the tire shape and strength) are laid out on the pattern.
Like HF said, the cords on a bias ply tire are laid at approximately a 45 degree angle to the beads. A set of cords going lower-right to uppper-left is overlapped with a set of cords going from lower-left to upper-right. This form gives the tire tremendous strength and rigidity, especially in the sidewalls. The form however tends to force the tread face to deform under load rather than the sidewalls.
A radial tire, however, has the cords laid out perpendicular to the beads. In other words, the cords goes straight from the left bead to the right bead. |--| This results in a less rigid tire that is more flexible. To control the shape of the tread face, manufactures will run "belts", either aramid or steel, on top of the cords parallel to the beads. Much like a ring on a finger. The belts keep the tread face from deforming under load which means that the sidewalls tend to flex more. The end result is that more tread remains on the road at any moment in time. The added benefit is that less energy is wasted in deforming the treadface which translates into less rolling resistence leading to better gas mileage.
2c in the jar, please.