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.
"Simple Switchers" (brand name) are easy to use, buck or boost, very efficient switch mode power supplies - boosting the output current should be easy, but I'll let the experts answer that, otherwise I'll get into hot water.
The simplest answer is to return the fan you bought, and get one intended to run from 9 V.
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.