I'm an amateur with no idea about most of it (see /u/tplgigo comment) but in case you don't know you've also got a fairly modern (D-link) network switch at the botom left, used to allow one network cable (coming from a router for example) to connect several network cables that can then connect to a number of wired devices. Maybe that was used for the rest of the home/upstairs.

I'd guess that the removed router and modem used to be near that Dlink switch (where the two black screws are), and that it may have been connected to a telephone line, could it be that small port on the far left? Can't tell if it's RJ-45 (network) or RJ11 (phone line) etc. though, or where it's going to. If you have a phone line working in other rooms, you could try pluggin an old phone into it and get a dialing tone? Then you could plug your own equipment in there?

Hope that helps. Hopefully someone will correct any errors I've made.