Minecraft wiki entry for Server, associated text: The default multiplayer software is free of charge and is available by Mojang Studios for Windows, macOS, and Unix-like systems (Linux, BSD...). See the Minecraft multiplayer server tutorial and Mojang Studios' Minecraft multiplayer server download page for help.
Opening a world to LAN provides a server that is accessible only to other people on the local network unless port forwarding is set up on the router. See the setting up a LAN world tutorial for more information.
Custom server software such as CraftBukkit, Spigot, and Sponge provide players a way to add plugins to a server. These are generally used for larger servers that run plugins to ensure griefer protection, non-vanilla commands (e.g./sethome), custom minigames, etc.
Rented servers are servers hosted externally by another company. These hosted servers are (usually) not provided free of charge - the player must rent them on a regular basis.
Realms are Mojang Studios' official hosting service that allows a limited number of players onto a server. Note that the Java Realms are still separate from Realms for other versions of Minecraft.
Dedicated Servers are servers that provide a wider range of power sources. These servers are much more flexible and allow for almost complete control.