The bombs went off in Afghanistan's crowded southern Manzalbagh district shortly after midday when many children were on their way home from school.
After the first bomb went off, curious onlookers went to the scene where nearly 20 minutes later the second more powerful explosion occurred.
"There are 15 dead and 58 injured," deputy spokesman for President Hamid Karzai, Hamid Elmi said.
Twelve children were among the dead and 40 were injured, he said.
Kandahar deputy police chief Salim Khan said one man was arrested in connection with the initial explosion, which caused no injuries, and another man was being held in relation to the second blast.
The bombings, which Karzai condemned as acts of "cruelty and barbarism", were the latest in a string of violent incidents in Kandahar, the former stronghold of Afghanistan's deposed Taliban rulers.
Tuesday's attack occurred outside a customs house and near the barracks of an Afghan commando unit working with US forces to hunt down Taliban and Al-Qaida fighters.
Meanwhile, Afghan authorities have arrested 11 people over a 28 December bombing near Kabul international airport which killed six people, including the city's intelligence chief.
"We have arrested 11 people, they are all Afghans trained in Pakistan," Kabul's deputy intelligence chief, Aman Allah Khan, said on Wednesday.
The blast killed Abd al-Jalal, head of the 21st intelligence directorate - a department of the Afghan intelligence service - four of his bodyguards and the bomber.
More than 400 people have been killed in the past five months in between remnants of the Taliban and pro-government forces.