Election commission chief Attahiru Jega said the delay until 28 March was necessary because of a lack of troops available to protect voters.
Nigeria’s military has been battling an insurgency by Boko Haram militants in the north-east of the country.
The opposition described the delay as a “setback for Nigerian democracy”.
Former military ruler Muhammadu Buhari, from the All Progressives Congress (APC) is challenging incumbent President Goodluck Jonathan, who heads the People’s Democratic Party (PDP).
Correspondents say it looks set to be a tight race between the two.
Mr Jega announced the delay late on Saturday.
He said it was necessary as he had been told that troops would not be available to help patrol the vote as they were too busy conducting operations against Boko Haram.
“The commission cannot lightly wave off the advice of the nation’s security chiefs,” said Mr Jega.
“Calling people to exercise their democratic rights in a situation where their security cannot be guaranteed is a most onerous responsibility.”
However, officials from the APC accuse the military of forcing the electoral commission into the delay to help the sitting president’s campaign.
APC chairman John Odigie-Oyegun said the “highly provocative” move was a “major setback for Nigerian democracy”.
Meanwhile, US Secretary of State John Kerry said the US was “deeply disappointed”.
“Political interference with the Independent National Electoral Commission is unacceptable, and it is critical that the government not use security concerns as a pretext for impeding the democratic process,” he said.
The ruling PDP party praised the postponement, and said the commission was facing “numerous logistical problems and numerous internal challenges”, AP news agency reported.
Analysis: Will Ross, BBC Nigeria correspondent, Lagos
The delay is highly contentious and will be seen by many Nigerians as foul play. What is not clear at this stage is whether it will favour President Goodluck Jonathan or his rival, Muhammadu Buhari.
Attahiru Jega made a point of saying this was a decision taken by the electoral commission but clearly the “referee” was under intense pressure.
The Boko Haram conflict has raged for five years but just days before the vote, Professor Jega was suddenly told the entire military would be focused solely on the north-east – in other words, “you are getting no help from the military, you are on your own”. With the threat of violence so real, he was put in a tight corner.
It seems highly unlikely that the conflict will be brought to an end within the next few weeks, so will the election be held at all? That may depend on whether some powerful personalities feel President Jonathan is well placed for a victory.
Who are Nigeria’s Boko Haram Islamists?
How have Boko Haram become so strong?
Parliamentary elections due to take place on 14 February have also been postponed to 28 March, and elections for state governors and assemblies slated for 28 February have been moved to 11 April.
Thousands of people have died as a result of the Boko Haram insurgency over the past six years, while more than a million people have been displaced.
The militants control a large stretch of land in north-eastern Nigeria but have also started attacking neighbouring countries.