The military will take “all necessary action” to “put an end to the impunity of insurgents and terrorists” in Borno, Adamawa, and Yobe, he said.
Mr Jonathan also ordered more troops to be sent to the north-eastern states.
Militants from Boko Haram have been blamed for most of the violence, which has left 2,000 people dead since 2010.
The Islamist group, whose name means “Western education is forbidden” in the local Hausa language, is fighting to overthrow the government and create an Islamic state in the north.
Analysis
President Goodluck Jonathan has declared a state of emergency across an already heavily militarised area of northern Nigeria.
This is not the first time he has taken such action following an upsurge in attacks by Islamist militants. The question is whether more troops on the ground will make any difference. Many analysts say the military has been losing the battle for hearts and minds and abuse allegations have helped boost Boko Haram.
Attacks have largely been hit-and-run or even suicide missions, but by saying areas have been “taken over by groups whose allegiance was to different flags and ideologies”, the president has made the somewhat embarrassing admission that the state no longer controls the entire territory of Nigeria.
Mr Jonathan also urged politicians in the affected states to co-operate with the armed forces and the police. This raises the question of what they have been doing. After all, some northern politicians have been accused of backing Boko Haram.
Nigeria – a multiethnic nation of more than 160 million people – is also affected by a spate of conflicts over land, religion and oil.
In the latest violence, 53 people were killed and 13 villages burnt in central Nigeria’s Benue state on Tuesday.
The conflict, which started last week, is said to have been caused by a dispute over land ownership between cattle herders and farmers.
‘We will hunt them down’
In a pre-recorded address broadcast on Tuesday, President Jonathan said: “What we are facing is not just militancy or criminality, but a rebellion and insurgency by terrorist groups which pose a very serious threat to national unity and territorial integrity.”
Referring to recent attacks on government buildings and killings of officials and other civilians, he said that “these actions amount to a declaration of war”.
“We will hunt them down, we will fish them out, and we will bring them to justice,” the president said.
“The chief of defence staff has been directed to immediately deploy more troops to these states for more effective internal security operations.
“The troops and other security agencies involved in these operations have orders to take all necessary action… to put an end to the impunity of insurgents and terrorists.”
Boko Haram appears to have gained ground recently, as Will Ross reports
At the same time, he stressed that – despite the state of emergency – politicians in the three states would remain in their posts.
The president has the power to sack local politicians and install a caretaker government in emergency circumstances.
“Already, some northern parts of Borno state have been taken over by groups whose allegiance are to different flags than Nigeria’s,” he said.
Later, a spokesman for the president, Doyin Okupe, said the governors of Borno, Yobe and Adamawa had been “very well briefed” and were “in full support” of the decision to declare a state of emergency.
“There’s a need for the government to step in and do the necessary to once and for all find a way out of this quagmire,” he told the BBC.
On Monday, the Nigeria Governors’ Forum, which represents the leaders of the country’s 36 states, warned Mr Jonathan against imposing emergency rule.
It is not the first time that the president has declared a state of emergency, but this is a clear admission that far from being weakened by the army offensive, the threat of the Islamist militants is growing, says the BBC’s Will Ross in Lagos.
And it is the first time that Mr Jonathan has admitted that parts of the country are no longer under central government control, says our correspondent.
Last week, Mr Jonathan had to cut short a trip to South Africa to deal with the growing violence.