Basically, all fans would work with only 2 pins. These 2 pins are the GND (Pin #1) and the +12VDC (Pin #2), very similar to the first two pins of a 4-pin Molex cable from your PSU. Supplying +12VDC power to this fan will just run it at full speed. Connecting a 2-pin fan to a 3-pin or 4-pin motherboard header might require some cable adapter. The motherboard header may be able to control the fan speed by changing the supplied voltage but it cannot monitor how fast or slow it is due to just having the 2 pins.
If the fan has 3 pins (such as yours based on your picture), then works similar to the 2-pin fan with just the addition of a 3rd pin that monitors how fast or slow it is. These 3 pins would then be the GND (Pin #1), +12VDC (Pin #2), and SENSE (Pin #3). The 3rd pin only acts to "report" the speed of the fan's rpm, a tachometer. This way, if you plug the 3-pin fan to a 3-pin motherboard header, the motherboard will be able to detect/display the speed of the fan and you can also control its speed by changing the supplied DC voltage. Hence, if you see a 3-pin header on a motherboard, it is usually referred to as a DC fan header.
In the case of your motherboard (Gigabyte B250M-DS3H), it has 2x 4-pin headers, namely the CPU_FAN and the SYS_FAN. As per your manual, both of these fan headers are PWM (Pulse Width Modulation) headers, due to the addition of the PWM pin (Pin #4). Note that not all 4-pin fan headers in a certain motherboard are PWM headers (some are 4-pin headers with the 4th pin having no electrical connection at all).
PWM fans (with 4 pins) can take advantage of these 4-pin PWM headers. The 4 pins are the GND (Pin #1), +12VDC (Pin #2), SENSE (Pin #3) and the PWM (Pin #4). The system is similar to the 3-pin headers but the 2nd Pin supplies a constant +12VDC and does not change. The fan motor itself has a PWM chip which uses that 4th pin to control the current being drawn by the fan, thus, changing its speed without lowering the +12VDC supplied.
3-pin fans can be installed directly to 4-pin motherboard headers, leaving Pin #4 (PWM), unplugged. The effect will be similar to plugging the 3-pin fan on a 3-pin header but the +12V supply in Pin #2 is non-variable. This means your 3-pin fan will always run at 100% and there is no way to control the fan speed.
In the same manner, 4-pin PWM fans can be installed directly to 3-pin motherboard headers too. But, the difference is, you can change the fan speed by adjusting the voltage in Pin #2. But the 4-pin PWM fan will not work as a true PWM system (where voltage is constant).
Now as for your splitter cable (I never knew that splitter was included in the Phanteks P400), it seems to be a 3-way 2-pin fan splitter cable (based on the picture, but, I cannot see the other end of that splitter in your photo). It can be plugged in directly to the PSU via a 2-pin-to-4-pin-Molex adapter, to which the system would work as I have described in the first paragraph above). If the other end of the 3-way fan splitter (which I can't see in the photo) is a 3-pin, then, you can plug it directly to your motherboard's SYS_FAN, but, it will only function similarly as if plugged in to a PSU (since your SYS_FAN is a PWM header supplying constant +12VDC).
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