A linear-phase equalizer uses linear-phase filters. This means that when a signal goes through the filter, all frequencies should experience the same time delay (known as “pure time delay”), which preserves the wave shape as much as possible.
All filters have phase shift. However, it is possible to design a filter so that the phase shift is linear with respect to frequency, and thus translates into a pure time delay. The phase shift doesn’t “color” the sound. A system that accurately preserves such relationships is said to be “phase linear.”