This blog will tell you what you need to know about hosting a server for Minecraft Beta! =)
Things you need:
* A computer (obviously >_> )
* A router
* The server software (get it from minecraft.net/download.jsp)
* A static IP (this blog is written assuming you already have one. googling "how to set up a static ip" is your best bet, it shouldn't be too hard to set up. I might write a blog about it if people are experiencing difficulties.)
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First off, porting forward. You'll want to take a look at your router and see the who the manufacturer and what the model is (this info isn't hard to find, it's usually on the back of the router. examples of manufacturers include Linksys and Netgear, and models are usually just letters and numbers.) Go to portforward.com, and find your router on their list. Look at the default guide for your router, and follow the instructions listed there. The IP you'll want to forward to (192.168.1.*, etc.) can be found by opening Command Prompt and putting in "ipconfig". (Windows only, not sure how to find it on a Mac, sorry!) You will want to put both the starting port and ending port as 25565, and the type as TCP/UDP. You can name the service whatever you like ("Minecraft" is always a good name). Log out of your router.
Second, place Minecraft_Server.exe/.jar somewhere and run it. The server console should come up. Now go to whatismyip.org, write down the number there, and give it to your friends (enter it in the "Multiplayer" menu). If you performed all of the steps correctly, they should be able to connect to your server!
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Questions/feedback in the comments, I will try to reply to most of your questions ASAP. (Also, hosting a Classic server is basically the same thing except with different software and a different method of connecting) Anyway, hope you liked the blog!