Response time is the amount of time a pixel in an LCD monitor takes to go from one value to another and back again. It is measured in milliseconds (ms). Lower numbers mean faster transitions and therefore fewer visible image artifacts.
Older monitors with long response times would create a smear or blur pattern around moving objects, making them unacceptable for moving video. Long response times can be annoying to a viewer depending on the type of data being displayed and how rapidly the image is changing or moving. Many current(which?) LCD monitor models have improved to the point that this is only seen with extreme contrasts.