Modes of operation
The access point, providing WiFi is usually called AP, while the client is called STA.
It's possible to configure your device as a repeater or as something I will call a wireless-wireless router, since I can't find a consensual name.
- The repeater will bridge (transmit all Ethernet packets between STA and AP sides), using the existing network infrastructure for DHCP, routing, ...
- The wireless-wireless router will route selected packets between the STA and the AP side, providing DHCP and NAT.
The wireless-wireless router will need more configuration, but will only transmit packet as needed.
Hardware compatibility
Some wireless adapter are not able to use STA and AP mode at the same time (some adapters even can't serve as AP).
If you've got an adapter at hand, iw list can be used to establish if it is able to do act as an client and access point at the same time. As said in this question, you must look for the valid interface combinations part of the iw list command.
About the single 802.11 a/b/g/ac radio
Since 802.11 b/g/n uses 2.4 GHz and 802.11 a/ac uses 5 GHz, the adapter will need at least 2 phy, and will probably need to be configured as 2 separate access point, from the Linux point of view.
About the software
They are plenty of dedicated Linux distribution targeting small computers and WiFi routers, here are some examples:
All the previously listed distributions have their own flag on superuser.
About the WRT name
Interestingly, a lot of router-targeted Linux distributions contain the WRT word, since they started as an alternative firmware for the venerable LinkSys WRT54G.