He died in a Dublin Hospital at 06:40 GMT on Friday.
Fr Reid, 82, a  member of the Redemptorist order, secretly acted as a conduit between the republican movement and the SDLP.
He was one of the witnesses who confirmed the decommissioning of IRA weapons.
In recent years, he was involved in talks with Basque nationalists seeking independence from Spain.
In 1988, Fr Reid was pictured praying over the bodies of Army corporals Derek Wood and David Howes who had been dragged from their car, beaten and shot by the IRA.
The corporals had inadvertently driven into the midst of a republican funeral in Belfast.
The photograph of the bloodstained priest crouched praying over one of the soldiers was one of the starkest images of the Northern Ireland Troubles.
A Ministry of Defence spokesman said: "Fr Alec Reid's intervention to administer last rites epitomised his enormous faith and strength of conviction.
"His comfort was given amidst the enormous fears and tension on that terrible day in March 1988."
Sinn Féin president Gerry Adams said Clonard monastery, Fr Reid's base in west Belfast during the Troubles, was "the cradle of the peace process".
He told RTÉ Radio he was with Fr Reid on Thursday night in St Vincent's Hospital, Dublin, and had been due to visit him again on Friday.
"What Alec Reid did was, he lived the gospel message. He developed a view which was contrary to the official view, that there had to be dialogue, and he was tenacious."
Northern Ireland Secretary of State Theresa Villiers, said: "I heard with sadness of the death of Fr Reid.
"We all owe a debt of gratitude to him for the role he played in the peace and reconciliation process in Northern Ireland."
Former SDLP leader John Hume said Fr Reid was "not simply a go between" in the early days of negotiating for peace.
"He was an active player in fighting for an end to violence," Mr Hume said.
"Without his courage, determination and utter selflessness, the road to peace in our region would have been much longer and much more difficult to traverse.
"While we mourn the loss of a great man, we must also celebrate the legacy of peace and an opportunity to reconcile our people that he gave to us. It is an opportunity we cannot afford to waste."
Michael D Higgins, the president of Ireland, said: "His ministering of the last rites to the two British corporals brutally killed in 1988 offered us an image of decency struggling to assert itself amidst brutality."
There were tears and tributes in west Belfast as news broke of the death of Fr Alec Reid.
"He should have got a Nobel Peace Prize," said one man. "But maybe he turned it down. The only prize he wanted was peace."
One elderly woman, tears streaming down her face, said: "He should be remembered with respect.
"He was someone who children should aspire to be like."
Fr Reid's funeral will take place at 12:00 GMT on Wednesday after requiem Mass in Clonard church in west Belfast.
Fr Alec Reid, who was a key figure in the Northern Ireland peace process, acting as a go-between between the IRA and politicians, has died.