I recently bought a 12V small computer fan(about 1"X1") and it was running rather slow when I checked it with a 9V. The package says that the fan requires 12V to run at an optimized speed, so is there a way/circuit that I can use/make that increases the voltage from 9 to 12?
As others have said, a 9V battery won't produce enough current to power a small computer fan. Using any type of step-up circuitry is only going to increase the current demand on the battery. Even if you could get it to work, you'd get only 3-4 hours runtime at most. The short answer is that a 9V battery just doesn't have enough juice to power a computer fan.
You can try putting the 9V battery and 2AA(1.5V) battery in series to produce 12V. Or you can try making a boost converter circuit which is a bit more complicated but will step-up or step-down your voltage in a circuit. You can learn more about boost-converters here.