When a channel has been equalized the frequency domain attributes of the signal at the input are faithfully reproduced at the output.
Early telephone systems used equalization to correct for the reduced level of high frequencies in long cables, typically using Zobel networks.
These kinds of equalizers can also be used to produce a circuit with a wider bandwidth than the standard telephone band of 300 Hz to 3.4 kHz.
Modern digital telephone systems have less trouble in the voice frequency range as only the local line to the subscriber now remains in analog format, but DSL circuits operating in the MHz range on those same wires may suffer severe attenuation distortion, which is dealt with by automatic equalization or by abandoning the worst frequencies.
In digital communications, the equalizer's purpose is to reduce intersymbol interference to allow recovery of the transmit symbols.