What are Fan Controllers???

The Corsair AF 140 and 120 are said to require a "standard 3-pin connector" on the mobo. I presume that means they are "3-pin fans". That means they are best used connected to a mobo SYS_FANn port with 3 pins so they can be controlled automatically by the mobo.

The Noctua fan is labeled "PWM". This means it is a "4-pin fan" and should be connected to a 4-pin mobo SYS_FANn port ideally for automatic control.

Almost all mobos these days have at least one (often 2 or 3) ports for case fans, called SYS_FAN ports. The mobo itself can do automatic control of a fan plugged into such a port. That means that it constantly checks the actual temperature at a sensor built into the mobo and changes the speed of the fans connected to that port to keep the case interior at the right temperature. This automatic control is the default setting in BIOS, but you have options usually to bypass automatic control and set fan speeds manually.

Ideally you connect one fan to one port. If you have more fans than ports with a particular pin configuration you can buy a Y-splitter adapter to allow two fans to connect to one port. I recommend not more than two fans per one port, however.

You can mis-match pin counts - the mechanical and electrical characteristics allow this - and it will sort of work. A 3-pin fan connected to a 4-pin port will always run at full speed - no automatic control. A 4-pin fan connected to a 3-pin port will work fine, with possible difficulty at start-up time that probably will cure itself. But ideally you do not mis-match.

The mobo will also have a specific port for the CPU cooling system, CPU_FAN. You should use this for automatic control of your CPU cooling, based on temperature actually inside the CPU chip. Do NOT power any other fan from this. Pay attention to the number of pins on the CPU_FAN port (usually 4-pin these days) and buy the correct fan type for that.

If your mobo has a special port called PWR_FAN, it is intended ONLY for use with a special set of wires on some Power Supply Units (PSU's) that look just like the wires from a 3-pin fan. Its only purpose is to allow the fan in the PSU to tell the mobo its speed, but many PSU's do NOT have these wires coming out.

Your mobo also will measure the speed of fans connected to its ports. If you want to know what those are, usually you can see them in the BIOS Setup screens, but it's more convenient to use a software utility usually included on the CD that comes with the mobo to display them on screen as you are running normally. Such tools usually also display the measured temperatures, and sometimes allow you to adjust the cooling control settings.

Instead of connecting your fans to the mobo and allowing it to control them automatically according to actual measure temperatures, you can buy and install a separate fan controller. It draws its power directly from the PSU and you plug your fans into it. It has controls on the front that allow you to set fan speeds manually. But it has no way to tell you the actual temperatures, so you have to figure out for yourself what fan speeds to set, and when to change them. Further, the fan speeds cannot be sent to the mobo this way, so you may not be able to see them unless your fan controller also includes speed display.