I'm a new linux user and I'm having problems while creating any directory. It says:
"mkdir: cannot create directory `Bill': Permission denied"

Any help ?

If you are trying to create a directory outside your /home/user directory, you usually need root/administrator permissions. The specific method varies a little depending upon which Linux you are using. Which Linux distribution of Linux are you using? and where (what directory) are you trying to create it in?

I am using Ubuntu 12.10 and I'm trying to create it in the "home" directory.

If you are creating a directory under /home/<your_user_account/ directory then it won't say permission denied. However, if you are trying to create directly under /home and then it will say permission denied because only root can do that. As you said that you are using Ubuntu I am assuming the account that you are using is the one that was created during the install. So instead of mkdir you can try using

Yeah, you're right. Now it works. Thank you very much

I think it's worth asking -- why are you creating a folder under /home?

It's my first time working on linux OS. So I'm just new to the whole idea of this system. That's why I tried to create a directory in the home (Which is the one that is open initially)

Ubuntu 12.10 has not been supported since May, 2014 so you won't have updates of software available. Using 12.04 which is supported until April, 2017 or 14.04 which is supported until April, 2019 might be a better idea if your hardware can handle it.

yancek makes a good point about using a supported version of Ubuntu.
If you want to create a directory to organise things you should be doing so in your own home directory which will be something like /home/bill and not straight in /home. The directory /home is where the individual home directories for users are created so unless you are doing so for a specific reason you should generally not create directories there.
Usually though when you open a terminal you start off in /home/bill and it's indicated by the prompt saying something like "bill@ubuntu:~$" where the "~" sign shows you are in your home directory and you can change to your home any time by typing "cd ~" without the quotes.

Okay. Got it !
You were very helpful. Thank you!