You're dealing with a limit of the way Windows works, and of a basic design element of the hardware. The latter point first.
In the design and construction of a mobo and other items, there has never been a standard route for taking an audio signal from somewhere and feeding it to a video output system. That includes both video generated by a chip in the mobo (which only started to show up in the late 1990's I believe) and video generated in a graphics card added into an expansion slot. When the new HDMI signalling system was designed, it was intended for much wider use than simply in computers, so it included the ability to carry stereo audio signals to the display device, anticipating that such devices might have their own speakers. That created a dilemma: how to get those audio signals from their generation point to the HDMI cable. On a mobo the solution was not too hard: just add connections on the board from the audio output chip to the HDMI cable socket on the rear panel. But for a video card in a PCIe slot, there was no standard way to send such analog signals from the mobo through PCIe slot contacts to the video card. So the solution adopted by almost all video card makers was to add a second audio chip to the video card, and have the Operating System (e.g. Windows) direct the digital info for audio output to that system on the video card, rather than to the audio chip on the mobo. Then the analog audio from that chip on the video card is simply connected to the appropriate pins on the HDMI output socket of the VIDEO CARD, not to the socket on the mobo's rear panel. So now your system with an added video card has TWO possible audio output chips.
Now, Windows has a limit related to this. No matter how many possible audio output devices your system has, Windows can only use ONE of these at a time. So it can direct digital signals to create analog audio output to either the chip on the mobo, or to the one on the video card, but not both simultaneously. Very commonly at the time that you first install a video card with an HDMI output on it (and hence containing its own audio chip), the card's Install routines install device drivers both for the video system and for the card's audio system. THEN it also goes into Windows configuration settings and sets them to use the VIDEO card's audio system, instead of any mobo-based system. Apparently that was done in your case, because you DO get audio out of your monitor's speakers, and that is fed from your video card's HDMI cable.
Because Windows is now set to send audio out through the video card, it will NOT try to generate any audio by the mobo chip that feeds its signals out of the mobo's rear panel jacks, and also the front panel headphone jacks. That's why you get no audio from any of those jacks.
YOU can change this any time you want to. Click at bottom left on the Windows Start icon, choose System ... Sounds. Look in there for a place you can set the Default Sound Output Device. (There also will be a similar item for Default Sound Recording Device, which can be set separately.) In there should be a chooser slot or window with options for your mobo system (usually Realtek) and your video card system (named for the card maker). It will be set now for the video card system. Simply choose instead the Realtek Audio system on your mobo and back out. Now ALL audio will come out of your mobo jacks (rear panel and front jacks), and none via the HDMI cable to your monitor. This does mean that your monitor will NOT give you any sound output. IF you need to have sound there, and IF your monitor has a separate jack for INPUT of audio, you can connect to that input and feed it from your computer's rear panel light green jack. Or, of course, you can just use your other speakers normally plugged into the rear panel jacks. OR, you can simply go into that Default Audio Output chooser and change it for whatever work you are doing.
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