Does it mean that you need 9 V to make the motor start, or is this the max voltage before it is damaged?
The ratings and specifications for small hobby motors are often not very well defined. The ratings for a DC motor with a permanent magnet field and a commutator are not very complicated. A DC motor that is advertised as a 9-volt motor should have torque, speed and current ratings based on operation at 9 volts. The torque rating should be the torque that it can safely provide to drive a load continuously or according to a specified duty cycle. The rated speed should be the speed of the motor when operating at rated voltage and rated torque load. When operating at rated conditions, the motor should draw the rated current from the power supply.
A small DC motor can be expected to operate over a range of voltages with speed proportional to the voltage applied, but also influenced by the load torque applied. Such a motor can probably operate at 20% or 30% above the rated voltage with no immediate harm, but increasing the speed will make it wear out more quickly. There will be some minimum voltage below which the motor may not start with with a load connected. Most motors require cooling by air motion generated by their operation. If they run too slowly, they may overheat.
The ratings for other types of motors are generally similar to DC motor ratings. Small brushless DC motors often have specifications based on the input to the electronic speed control (ESC) that they require. Some small DC servo motors are sold with an ESC built into the motor housing. Small fans used to cool computers are often brushless DC motors with a built-in ESC.
AC motor have specifications that are similar to DC motors, but the operating frequency determines the speed and operation with variable voltage is very limited. Most AC motors are rated for operation at 120 volts and higher. There may be a few designed for as low as 24 volts.
9V motor is a motor that uses the nominal voltage of 9 volt.
The motor might start spinning at 3V. If you use it at 12V it will burn out (only bad smell and if you are lucky some sparks). At 12V the voltage is higher but the current will also be higher, so the watts absorbed by your motor is almost double (180%).
\$P=U\times I\$ (Watts=voltage*amps)