How can I host my own Minecraft server?

Updated answer as of 11/5/17

Here is the updated link to the tutorial on setting up a server on the wiki.

You can download the latest jar file from minecraft.net. It may download as a zip extension, in which case, you can change it to jar.

If you are interested in an alternative server which allows more configuration and the use of plugins, you can try bukkit, spigot, or paper spigot. Make sure to get the latest build for the version you play.

Setup of alternative servers is very similar to vanilla servers but adds more that can be done with plugins and more configuration files. Here is the wiki on setting up a Spigot server.

You can also use a single player world and Open to LAN.

The basics for starting the server are

1. Put the jar file into a folder(as it will create files).
2. Run the jar file. It will run and shutdown after creating files.
3. Open eula.txt and change eula=false to eula=true and save.
4. Open server.properties and configure to your liking.
5. Run jar file again. It will create the world files.

You can now log in to the server. From the computer the server is running on you can use the IP:localhost to verify the server is running properly.

From here, it can be logged into from any computer on the network using the internal IP of the computer running the server. On some systems, you can use the name of that computer as the server IP address.

If you want to go online, you will have to port forward your router. Then other people will log in to your server using your external IP address. You can use whatismyip.com to obtain that IP address. You can also use a dynamic dns. This makes it so you don't have to give out an IP number but instead a web address which may be easier to remember. It also allows the server to change external IP numbers without requiring external player from having to go change the IP in their multiplayer list. Currently noip.com offers this service for free.

Here is a tool to see if your computer is up to the task of running a server for others to play on.

Another option is to use a server hosting service. This allows you to run the server on another computer. Several offer protection from DDoS attacks which is recommended if the server is to be public. Some offer subdomains which give the benefits of the dynamic dns.
Here are a few: mcprohosting.com, playpro.com, and currently server.pro even has a free option.

This is a rather old post but the other answers seem to have missed the obvious.

You are on a LAN network as you say, so you are both connected to the same router when playing. The one of you to host the server simply needs to create a new single-player world, load it up, open the menu, click "Open to LAN", choose the desired settings and your other half can join from the multiplayer menu at the bottom under "LAN Worlds". You will need to re-host the world every time you wish to play.

It is also possible to set one up Hamachi

The benefits of this is that,

• No need to open ports,
• Closed server so no people will come on and grief without being on your hamachi
• It's free and easy to get setup and running.

Cons are

• Can be limited to number of people on it.
• Higher encryption so can be slower.
• Requires other people to download and install hamchi