Another person with whom you share a deep, unshakable emotional, intellectual, physical and spiritual connection. Simpler phrases like "on the same wavelength" or "cut from the same cloth" are vastly insufficient.
You are both, in all the ways that matter, the same person. Once experiencing this epiphany, you will not only come to truly understand another's soul, but better understand your own as you are merely reflections of the same inner being.
We are imbued with unique qualities at birth/conception/creation that follow us and develop our whole lives, some inner resonance. And somewhere, throughout that life, if we're fortunate and have our eyes, ears (and hearts and minds) open, we'll come across someone who shimmers the same way we do and be elevated, forever changed.
There are many people in this life whom we may "get along with" or think similarly to. Yet with one's soulmate there is an striking additional component that is not easily defined. A sensed shared "vibration" that at times just levels you with merely a glance or being physically near that person.