Trying to make sense of setting up wired internet throughout house

[Bitterslol]

So with a few smart TVs and desktop computers throughout the house, I want to utilize the Cat 5 outlets in each room (the outlets typically have a blue colored ethernet port, and 2 white colored ethernet ports with one cable line...if that matters) to run wired internet to each device. In my mechanical room, I have this setup/box but can't seem to figure out how to apply my router (the model sets up easily, works and the router puts out wireless signal - just need to figure out which cables to plug in/etc).

Pics to follow (sorry, forums won't let me give the direct link): imgur.com *** /a/HdQPh *** (drop the asteriks)

The grey cables and blue cables on the left are all labelled (for which room they represent), but the grey cables seem to be missing some rooms or have had the cable cut/terminated. Also, only one of those grey cables plugs smoothly into the router, the others ones don't 'snap' into the ethernet port. Are the blue cables going into some type of switch that I would just plug my router into? Let me know if I can provide any more information.

I really appreciate any help, thanks in advance.

[lockblock]

Looks like a patch panel to me. Unless you see some sort of ac adapter going to one of them it is certainly a patch panel. You should be able to plug them all into a router or a large switch which would be connected to the router.
You could try bending the clips of the grey ethernet cords outwards a little bit to get them to snap in properly. (they will snap off if you bend too far or too much)

[Tradewind]

It's a Voice/Data distribution panel or TLDM. In short some of them are Telephony and some are Data. However, from looking at it (the photos are kinda balls :P) those all appear to me, to be setup for Voice only. That dangle of grey cat5e's you have hanging, are possibly your data ports on the wall plates.

The larger block on the right is where your phone lines come in from the demarc, into the lanes below, it's then patched over to the smaller tldm strip on the left. Hence my suspicion that they're all just telephone.

The small bundle of flat grey cat5 cables could be used to feed data to their respective wall plates too, but...it might also kill your phones From there instead of patching those flat grey cables back to the main tldm you would just put them right into a switch/router. But again, phones.