You can host the java version of the server on any machine.
You can also just expose any client's game instance to your local LAN - that's all I do when playing with my grandkids.
There are ways of doing multiplayer on your local LAN with a single account as well by simply editing your name in a single config file per machine. Google is your friend there. :)
There are basically two version of Minecraft at this point. The "classic" Java version, and the MCPE version, which is what you get on Apple devices, Android and the "native" Windows 10 version.
The latter has "realms", where you can just pay to have Microsoft host it for you, and is tied into Xbox Live.
The official names of the different versions are "Java Edition" and "Bedrock Edition".
Java Edition is the one that lets you run your own server, is more moddable, and works on Windows, Mac, and Linux.
Bedrock Edition is the one that works on Windows, consoles, and mobile. People can invite people into their own world as long as they're online. Your only private server option is Realms, which is a subscription service. (Edit: Apparently there is finally publicly-available server software for Bedrock Edition. I doubt it's very moddable though, so don't expect to be able to run a public server with all the modded tools you'd expect if you were familiar with running Java Edition servers. Also, I don't think the console versions of Minecraft can join arbitrary servers.)
The Java Edition and Bedrock edition are not cross-compatible with each other. They are fully compatible within each-other though. Bedrock Edition supports cross-play between different consoles and PC, etc.
you can run a server at home, also the java client can run as the server so if people are on the same LAN they can connect directly to whoever opens their world to LAN. If you want to play with friends or relatives remotely then the easiest, but not the cheapest, option is Minecraft Realms which allows you to control the player access list.
