Get a VPS
In the long run you should use a VPS instead of hosting on your own computer. This way you can turn of your computer at home and still be able host the server 24/7.
- The server should have at least one gigabyte ram.
- The server should not use OpenVZ for virtualization. It would work but you might face memory allocation issues. You should go for Xen virtualization.
In the continuation I will assume the VPS is using Ubuntu as OS.
sudo apt-get update sudo apt-get install sudo apt-get install openjdk-7-jdk
That will install java for you.
Get the minecraft server
You should get to know the terminal command screen if you does not already.
sudo apt-get update sudo apt-get install screen
Now you can start the server in a screen and detach from it to keep the server running.