Hi, not much of a professional when it comes to this topic, but if I understand your goals correctly you are going to need a back-end / server-infrastructure.

For creating a client-server or a peer-to-peer game nothing "extra" is required, you can simply use the built-in networking api in Unity (as I know the new one is called Unet, but there is a legacy one if you use an older version). This will allow you creating a game where multiple players can play together over network.

For storing accounts, their data (like weapons/items bound to accounts), authenticating users and for registering dedicated servers or players hosting games, you are going to need a server machine and a back-end service/software. This is a whole different animal type than both the client application or the game server code/application, and I guess a lot of work if you don't really know what you are doing :(.

I would most probably go for the first goal at the beginning, and stack up knowledge about the second as I go, or find someone who is capable and can develop the second part, but if you insist on going forward, you should look into following technologies:

Database tech, SQL comes to mind (free SQL server: MySQL)
Back-end and server tech, maybe in your case this could be done using a web-server with a language like php (Ruby and Python is a common choice too), or you could develop a dedicated server application accessing your database using Java or C++.
You are going to need hosting too for your back-end, a virtual machine or a dedicated server machine or some kind of cloud solution.

Again, not proficient in this area, but as the general steps, something like this is correct...