Access point or Range extender confusion

Here's my situation. I have a home that also has a small casita out back that I use as my office. In my home I have  Nighthawk RAX50 router that my cable modem is attached to. WiFi works fine throughout my home but it doesn't reach the casita. I do happen to have a CAT6e line running from my router to the casita and it is plugged directly into my desktop PC which works great since it is hardwired. But that still leaves me with no WiFi in the casita.

I want to keep my office PC hardwired but I also want to have wireless in the office for my phone/tablet and guests. 

I'm confused about what I need, an access point or range extender? 

Ideally I would plug the one CAT6e line into a "box" that then gives me a hardwired output to my PC and potentially other hardwired devices and WiFi in the rest of the office. I want the WiFi to be the same so when I walk from the office to the house it stays on the same network.

Thanks for the advice.