When exactly does an address of the form http://:3838/ go from being local to being on the internet?

I'm very new to server administration, and I'm trying to deploy a web application (in particular, an R Shiny web app) from behind my company's firewall.

I've succeeded in getting the app set up such that any computer on the same network as the hosting server can enter http://<server-address>:3838/sample-apps/hello/ into their browser to access the app. (I used a program called Shiny Server for this, which runs on any Linux server but is only supported for Ubuntu)

My confusion lies with how to extend this so that machines not on the network can also access this page. Which leads me to the question:

When exactly is something like http://<server-address>:3838/ only accessible locally? I've had to access webpages like this before, i.e. with just an IP address, but sometimes the IP was something on my local network and sometimes the IP was something on the internet. How exactly do I set up the latter situation?

(P.S. the default port is 3838 but that's not vital to my question I suppose, I should've left it out of the title)