The militants opened fire on residents, set fire to houses, and targeted a crowd with suicide bombers near the northeastern city of Maiduguri.
Around 30 people were reported killed and 80 injured when blasts from three suicide bombers hit the Chadian island of Koulfoua in Lake Chad.
The 15-day state of emergency will allow authorities to increase security, impose a curfew, and restrict the movement of goods and people, according to an announcement on state television.
The militant group released footage that purportedly shows a Nigerian soldier being decapitated on the same day that coordinated explosions killed 37 people.
The president claimed that Boko Haram has a new leader, and said the fight against the Nigerian militant group would be over "before the end of the year."
Islamist militants have multiplied deadly attacks in the region, often using women and girls concealing explosives under the religious veil.
Boko Haram fighters are hiding on hundreds of small, inaccessible islands scattered across Lake Chad, forcing thousands to seek shelter in a refugee camp where living conditions are extreme.
Boko Haram fighters reportedly seized control of a town in northern Nigeria on Friday, just as the Islamic State released images praising the militant group.
Witnesses reportedly said a girl who killed herself and at least five others in an attack on a market in northeastern Nigeria could not have been more than eight years old.