The anterior (or ventral) horns refer to the part of the spinal cord where the motor neurons course downward from the brain, eventually on to the muscles they control. When they go though the anterior horns, these neurons are referred to as lower motor neurons.
So anterior horn cell disease includes ALS (which also involves the lateral corticospinal tracts in addition to the anterior horns), progressive muscle atrophy, and any other motor neuron disease which has lower motor neuron involvement. Pure lateral sclerosis is an upper motor neuron disease without anterior horn cell involvement and would not be in this category. Interestingly, polio might also technically be considered an anterior horn cell disease but is not lumped in to the motor neuron disease group.
Since ALS and other motor neuron diseases are diagnosed by a variety of clinical criteria, including the EMG, the person reading the EMG might refer to “evidence of anterior horn cell disease”. To diagnose ALS, these EMG findings would need to present along with other clinical features of upper and lower motor neuron disease on exam and ruling out other clinical conditions.