The "crossing" was a term used by the non-corporeal lifeforms known as "Wisps," referring to the physical act of evicting the consciousness from a physical entity's body, and subsequently taking control of that entity.
In 2152, the crew of the Starfleet vessel Enterprise NX-01 encountered the Wisps, one of which performed the crossing with Commander Charles Tucker III. While existing in a non-corporeal state, Tucker reported a series of experiences, including swimming in Tarpon Springs with his old girlfriend Lisa, riding with Hopalong Cassidy, and snow-sailing in New Zealand. According to Tucker, the Wisps were attempting to initiate a cultural exchange, wherein they could experience life as did their ancestors, and the Human crew of Enterprise could learn what it was like to exist as non-corporeal lifeforms.
It was soon discovered, however, that the Wisps' true intentions were to permanently take over the bodies of the Enterprise crew, and subsequently move on to another vessel of corporeal lifeforms to perform the same function. With assistance from Doctor Phlox, whose Denobulan physiology was resistant to possession by the lifeforms, Captain Archer was able to initiate a plan whereby the affected crew were rendered unconscious by exposure to high levels of carbon dioxide, forcing the Wisps to vacate their bodies. (ENT: "The Crossing")