In Tuesday's first incident, a Toyota 4x4 vehicle with two people inside detonated a cache of explosives on the road from Herat International Airport to the Guzara district of western Herat province, according to the Afghan interior ministry.
Al Jazeera's Bernard Smith, reporting from Kabul, said the vehicle drove through a checkpoint on a bridge 2km ahead of its target and ignored two orders to stop before the explosives were detonated.
"The explosion was so strong there are causalities among police and civilians", said Nasar Ahmad Popul, the Guzara administrative chief, who was inside the building at the time of the attack.
Mohiddin Noori, spokesman for Herat's governor, said six civilians and three policeman were among the nine reported deaths.
From the perspective of one neighbourhood in Herat
"The number of casualties may rise," Noori said of the attack.
The Taliban claimed responsibility, but said a police bullet led to the detenonation that led to civilian lives.
The building, home to the offices of Guzara's police chief and administrative chief, also suffered damage, Popul said
The attack took place as people were gathered outside the building, waiting to see government officials about various matters inside.
A reporter for the AFP news agency on the scene said he saw shattered bodies lying among rubble and mangled pieces of metal after the bombing.
The responsibility for nearly all of Herat province has been transferred, or is in the process of being transferred, from NATO troops to Afghan forces.
In the day's second attack, eight police officers were killed after three suicide bombers targeted police headquarters in the southern province of Helmand, local security sources told Al Jazeera.
Two of the bombers were able to blow themselves up inside the old police headquarters in Musa Qala. A third bomber was shot before his vest could be detonated, the sources said.
Another 12 people were injured in the attack in the building that now serves as accomodation for police officers and the chief of police's family.
The Taliban claimed responsibility for the the Helmand attack too.