The explosion at a boys’ school in the town is believed to have been caused by a suicide bomber dressed as a student.
The militant group Boko Haram is believed to have carried out the attack, police said.
The group has targeted schools during a deadly five-year insurgency aimed at establishing an Islamic state.
It is waging a sustained campaign to prevent children from going to school. It believes girls should not attend school and boys should only receive an Islamic education.
The explosion ripped through the assembly hall at the Government Science Secondary School, reports say.
Police spokesman Emmanuel Ojukwu told the BBC Hausa service the attack had left 47 people dead, including the suicide bomber. Another 79 were wounded.
“At about 08:00am [07:00 GMT], a suicide bomber disguised himself as one of the male students and while the school was holding its normal assembly, the bomb went off,” Mr Ojukwu said.
He added that police were investigating the explosion.
One student told the BBC he saw the mutilated bodies of fellow students at the scene, where emergency operations were ongoing. A resident reported seeing parents wailing at the sight of their children’s bodies at the hospital.
Soldiers who attended the site of the explosion were met with fury by the assembled crowds who pelted them with stones and accused them of not doing enough to halt Boko Haram’s insurgency.
A grieving relative told the BBC: “My brother, a student in the school, died in the blast. He was about 16 years old… We buried him at about 11:00am [10:00 GMT] today.”
“The government needs to be more serious about the fight against Boko Haram because it is getting out of control,” he added.
Schools in Yobe state have been frequently attacked by Boko Haram militants.
The state is one of three in Nigeria that have been placed under a state of emergency as a result of the group’s activities.
Potiskum, one of the largest towns in Yobe, has been targeted before by Boko Haram.
Last week, a suicide bombing killed 15 people in the town.
The bomber joined a religious procession of the rival Shia Muslim sect, before blowing himself up.
In April, Boko Haram sparked global outrage by abducting more than 200 girls from a boarding school in Chibok town in Borno state.
Boko Haram leader Abubakar Shekau has dismissed government claims to have agreed a ceasefire, under which the girls would be released.
He says the children have converted to Islam, are learning to memorise the Koran and have been married off.
Credit: The BBC