The Order said Mr Nelson had been been ill for a short period of time.
Mr Nelson, from Dromore, County Down, was first elected to office within the Order in 2004.
Edward Stevenson, the Order's grand master in Ireland, said the institution had "lost not only its heartbeat but perhaps its greatest ever advocate".
"Drew was a towering figure within the Orange fraternity, whose commitment and devotion to the principles of Orangeism are simply unrivalled.
"No other individual, over the course of the past decade, has done more to champion the cause of the institution and its membership.
"His contribution to promoting a tradition he was immensely proud to represent was truly outstanding."
Ulster Unionist MLA Danny Kennedy said Mr Nelson led change in the Order that "saw its image and reputation improve significantly across the world".
"I knew Drew primarily through my membership of the Orange Institution, and his commitment to the Orange Order and his desire to improve life in Northern Ireland was obvious."
DUP MLA Peter Weir, who was a close friend of Mr Nelson, said he was "really sad to learn of the death" of his good friend. He described his death as a "massive loss to the Orange Order, but especially to all of us who are his friends and family".
TUV leader Jim Allister also said he was "deeply saddened" to learn of his passing.
"Drew has fought a valiant battle with cancer for a number of months, a trial he faced with tremendous courage," he added.
Drew Nelson, the grand secretary of the Orange Order, has died at the age of 60.