Delay is the time from beginning transmitting to receiving of signal or message.
Delay spread is the increase of the propagation delay due to the signal travels in different ways, such as in mobile communication, signal transmission to a receiver in many ways due to reflection, diffraction, refraction … See more in multipath propagation.
Assuming, the transmitter transmits a very narrow pulse, at the receiver, because of multipath propagation, the receiver receives a group of pulses like in a below image. ∆D is delay spread.
Note that, delay spread occurs not only in the mobile communication systems, it is in all radio (wireless) systems. However, in many cases, it is too small.
Delay spread in television causes shadow zones (analog television). In optical communication, so light in multimode fiber transmissions in many different rays (if explained by the optical method) will expand pulses in output. In nature, it is delay spread.