Your photos show an unused DVI video output connector, and your blue cable plugged into the socket next to that, but I cannot read the label on the socket in use. Does it say HDMI? Or perhaps VGA? I will note that the blue connector in there is a locking type (with thumbscrews) widely used for VGA cables, but I rarely see that on an HDMI cable. Just so you know in advance, neither a VGA nor a DVI video output ever could carry SOUND on the same cable, so video cards that had ONLY those types of outputs normally do NOT have any audio capability, and my post above is not relevant.
The wider view shows me a normal-looking rear panel with connections from a mobo, and NO cards of any kind added into a PCIe slot and providing output connectors from a slot - all four are still covered. It DOES show a group of six 3.5 mm female sockets for audio signals. That's where you can plug in stuff, depending on what things you have. In most cases these are 2-channel sockets so they handle stereo signals, or signals for 2 different units. The common colour code for those is:
Lt Blu = Line In
Lime Green = Line Out for Front stereo speakers
Pink = Mic In
Orange = Front Centre and Sub-Woofer
Black = Side Speakers Out
Grey = Rear Speakers Out
All this looks like your only audio output device is a chip on the mobo, probably by Realtek. Now normally, in addition to those sockets on the rear panel, there is a header on the MOBO for a cable to the front panel for Audio functions there. The Front Panel has two 3.5mm sockets for plugging in a headphone and mic set. The mobo header is male, with 2 rows of 5 pins but one pin missing. I suggest you check whether that cable is connected. With your system OFF, open the side panel. IF you have a mobo drawing, look for where that Audio header is. If not, look at all the labels on the headers and see if you can find it. Normally you cannot reach the other end of that cable to check its connection to the front sockets.
And here's another item you can try. Normally you'd use the front panel light green socket for the earphones. But you CAN use the REAR panel light green for that as an alternative.
Oh, another important thing to check, although this may NOT apply to you. On your headphone set, look closely at the jack on the end of the cable. If there really are TWO wires, each ending in a male jack, you're probably OK on this item. Each of those will have three contacts - Tip, Ring, and Sleeve. BUT if yours has ONE jack on the end, and that has FOUR contacts - Tip, Ring1, Ring2, and Sleeve - then you have a connector that cannot work in a normal computer output socket. You would need an adapter for that.
Now, here's an important item. Windows has a way to choose which audio output device (mobo chip normally) is used for its Default Sound Output Device. (This is not the same as speaker or headphone.) This is for systems that do have more that one way to do it. But your system may not yet be set to use what it has. At lower left type sound settings in the search box and click on the list to get a configuration box. The top section is Output. Click on Device Properties, and in that click Additional Device Properties. The General Tab should show Realtek Audio in the middle as the controller. Ensure that the box at the bottom is ENABLED. Note the diagram of rear sockets. Choose the Levels tab at top. There you can set the volume settings for the various outputs and inputs. On the Advanced tab you can click on a TEST button to get sounds (IF it's working!). Use Apply to back out.
Now go down to the Related Settings and choose Sound Control Panel. On the Payback Tab choose the Speakers, then Configure at bottom. You can check there which speaker system you have: 2-channel stereo (including earphones), Quad, 5.1 or 7.1 speakers. IF you have multiple speakers, there's a place to test each individually, as well as an overall test. You can go though a few more steps to make custom settings. The Recording tab allows you to choose the device you record from. When done you can Apply.
I note also that if you use the search box for headphones, it will pop up your browser with related items including several videos on how to fix headphones that won't work in Win 10.
While you have your case open, note down the maker and exact model number (or name) of your mobo so we can look up its manual, and post that here.