"Computer fans" come in two major types. The first type only needs two wires to run, a voltage wire and a return (ground) wire. The voltage required is design dependent, but today is usually +12VDC. If the fan has a label affixed, it may specify the max voltage, which would correspond to max fan RPM. This fan may or may not have a thermistor attached to sense the air temperature and control the fan speed (implies max voltage, i.e. +12V is always applied to the voltage wire). If no thermistor, then the fan speed is controlled by the "computer" and the voltage is lowered to run the fan at a lower speed.
Computer designers want to run the fan at as low an RPM as possible for reduced acoustic noise levels, but within the temperature profile required for computer / component / processor reliable operation. Therefore, a trade off is made so that the computer operating in a cool environment will not need to pump as much air through the system as a computer operated in a warmer environment. The fan can run slower, and the noise level (as perceived by the user) will be less.
A drawback to the two wire fan solution is that if the fan were to stop rotating (for whatever reason) the computer will quickly overheat and shutdown without much warning. So, a third wire was incorporated into the fan design to provide a "tach" feedback signal to the "thermal management" functions of the system.
A second fan type also uses three wires, and is completely different from the three wire "tach" solution. We still need the gnd wire, but the "voltage" signal is now a PWM (pulse width modulation) signal. In this case, the voltage is held constant, and the voltage is pulsed on / off. The "width" of the ON time vs OFF time controls the fan speed. The third wire used in this case is also the fan tach feedback signal.
If you are able to run the fan by just applying +12V, then you have the first type of fan. You do not need to connect the fan tack feedback signal wire.
Is this too much information??