Russian President Vladimir Putin, who is now in the Black Sea resort of Sochi, said in televised comments that a "tragic event" had happened and that "as a result of an explosion of an explosive device, people died and many are injured".
A local resident said officials had instructed parents to collect their children from schools and kindergartens in the city for their safety.
They said the attack had started with an explosion, followed by more blasts, and a hail of gunfire. Russian explosives experts later inspected the school for other possible bombs. Russia's Investigative Committee, the nation's top investigative agency, said the device that went off at the college's canteen was rigged with shrapnel.
The bomb blast took place in the city of Kerch, which is the entry point for a new 19km bridge linking Crimea to mainland Russian Federation. One witness told local television that armed gunmen were also involved in the attack.
An "unidentified explosive device" detonated at the college in Kerch in eastern Crimea.
Investigators, however, later identified the attacker as 18-year-old Vladislav Roslyakov, a fourth-year whose body was found "with a gunshot wound" on the premises. "I would have been dead now", she said.
Russian Federation annexed the peninsula in March 2014, after a referendum in which the population of Crimea chose to become part of Russian Federation.
President Vladimir Putin ordered the Federal Security Service to investigate the explosion, his spokesman, Dmitry Peskov, said.
It said in a statement that 13 people had been killed and around 50 wounded, many of whom were teenagers. 5-10 minutes after I left some people broke in, blew up the hall - all windows were shattered.
"There are already lots of people in the emergency room, and in the operating theater", a school employee told Tass.
According to Crimea Head Sergei Aksyonov, the death toll from the attack has reached 18, while more than 40 people were injured.