When something resonates, it will vibrate or echo in harmony. In the context of the song from Evanescence, the singer is saying that two people are in sync with one another, and that they move as one, breathe as one, and think as one. It's a very romantic sentiment, and used often in songs, poems, and stories about true love.
Typically, resonate is a word most commonly used in science, for example when talking about frequencies. You might also be familiar with resonance as part of the long form name of an MRI- magnetic resonance imager, a medical scanner.
In new age spiritualism, resonance is used to describe the harmony of energy flows in the human body, and some people believe that crystals can help balance the resonances within your body's energy lines.