Kevlar is able to stop a bullet due to its molecular structure. It is a light, polyarylamide plastic fabric, which has a high tensile strength. this means it takes a huge amount of energy to make its fibres stretch even a little. Each Kevlar molecule looks like a long twisting coil. During polymerisation these coils of molecules become tangled, causing it to be hard to stretch.
Inside a bulletproof vest are many strips and layers of Kevlar. When a bullet hits the vest, it tries to force it through the layers, but to do this it must push the fibres apart. The fibres are woven and resist this very effectively. The movement is translated into a stretching force on the fibres. Some will break, but most will absorb the energy of the bullet by stretching a small amount.
Nathaniel Marten, Science Museum