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.
'Devastating attack' 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.
By setting off the bomb during the morning assembly, the militants clearly aimed to kill as many students as possible.
Few of the attacks here are ever claimed by any group but Boko Haram will once again be suspected. The jihadists have carried out particularly brutal attacks on schools before.
Chibok is known in many parts of the world because of April's mass abduction of girls from that remote village. But there have been many other horrific attacks on schools which have received less attention - including last February's raid on Buni Yadi, in Yobe State.
Dozens of boys were burnt to death, shot or killed with knives in the dormitory. Female students were spared but told to never attend school again, go off and get married. Boko Haram wants the education of boys to be limited to strict Koranic studies only.
The insecurity in the north-east is so rampant, with entire towns and villages now in the jihadists' hands, it will be extremely hard for other bombings to be prevented.
"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.
Potiskum is no stranger to attacks - last week a suicide bombing there targeted Shia Muslims
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.