I would like to know if there is some way to run a hardline to a wirelss router and send the signal to something that converts the signal back to a hardline again other than using a networking adapter check attachment

I always wondered if you could have 2 wireless routers do that, which is:
Have Laptop 1 plugged into router 1, computer 2 into router 2, and have the wireless routers provide the link.
And connect to the Internet?

I'm not sure what you mean by "network adapter check attachment". The "whatever" device in your illustration could be another wireless router, a wireless bridge (example Amazon.com: Linksys WET11 Wireless Ethernet Bridge), or a wireless ethernet to serial adapter/media converter/device server gadget (example Digi RS-232/422/485 to 802.11b wireless converter) depending on what the device is you are trying to integrate. If this is in an industrial setting, many automation vendors have the latter type of devices for a variety of different media and interfaces in a DIN mount package for easy "wireless-izing" of equipment and machine tools.

I have a wireless access point/repeater that will do as you ask.

It's set up as a repeater, but the Ethernet ports (believe there are 4) can be used as wired connections. Forget the mode. Made by Dlink as an access point/bridge/repeater combo device.

Kiss, so basically, you could have 2 Laptops, neither wireless, and have each computer plug into its own wireless router, and they could connect with each other, and 1 could be connected to the internet, and the other through this "wireless bridge"?

Yes, but your version is too complicated.

Yes you can bridge wired to wireless to wired connections. Going from one media to anoter is typically called a "bridge"

An access poin (AP)t, or more typically a "wireless access point" allows you to "TAP" into the wireless network. Tapping in can be wired or wireless.

It's a network, so everyone (two laptops) can see each other and the (Router) proviides NAT (Network address Translation), so that each PC has a way to the internet.

You could use one for each, or two separate AP's To access the network.

There is only one set of incoming ports on the public side of the internet and these CAN be directed to a particular machine my modifying tables in the router.

No, I didn't know how to do it, I was also curious, thanks.