DC (direct-current) burn is a failure mode where the voice coil is burnt only at one end. This is an indication that it has been traveling in one direction more than the other. Since the transfer of heat is from the voice coil to the adjacent magnet and metal parts, the voice coil will be burnt on the end that stays the farthest away from the top plate.

Woofers: When a DC burn pattern appears on the voice coil of a woofer, the problem will be due to a fault in the associated electronic equipment. Most likely, the power amplifier has leaky or shorted transistors that are allowing its internal power supply voltages to be applied directly to the loudspeaker or loudspeaker system.

Midrange and Tweeter: When a DC burn pattern is observed on the voice coil of such devices, it DOES NOT always mean that the amplifier is faulty. In systems with passive crossovers, mid and high frequency drivers are protected from DC by the cross-over. The most likely cause of DC-like burns is an overdriven amplifier.
When an amplifier receives an input signal capable of driving it beyond its power rating, the result is clipping. This means that the negative and positive peaks of the amplifier's output signal are "clipped" off. The amplifier may also clip in an asymmetrical fashion, meaning that the positive side of the signal is clipped more than the negative (or vice versa). When subjected to an asymmetrical clipped waveform, one end of the loudspeaker's voice coil is "on average" spending more time outside of the gap (corresponding to the direction that is clipped) than the other. The end of the coil that is spending more time outside of the gap has poor heat transfer to the magnet structure. As a result, it overheats and burns.