There are many ways you can install Fedora on your machine.
I will detail one quick method.
a) Go to fedoraproject.org and download a Live-CD ISO that you prefer. They come in 32-bit and 64-bit versions. If you do not know if your computer is 64-bit, it is OK to then use the 32-bit version.
b) There are various LiveCD ISOs you can get to. I would recommend that you choose the 32-bit Fedora 15 Desktop Edition.
c) Once you've downloaded it, you will see a file called Fedora-15-i686-Live-Desktop.iso. If your computer has a cd writer, you can then use the cd-writing tool to write the ISO to a brand new CD. You are transferring an IMAGE not a file. Be careful when you using the cd writing tool that you tell the tool that you are writing an image not a file.
d) If you do not have a cd writer, and if you are running windows on your computer, you can download a liveusb-creator application to transfer the ISO to a USB flash device. See https://fedorahosted.org/liveusb-creator/.
Once you've gotten the CD or the USB ready, you can then boot up your computer from either the CD or the USB and you will be presented with a Fedora desktop.
From that point, you can then choose to install Fedora directly to your harddisk. NOTE: This WILL REMOVE whatever you've installed on your harddisk.
Now, if instead of installing on you INTERNAL hard disk, you can install it on an EXTERNAL USB harddisk. For this, you will have to click on the "Activities" menu option at the top level hand corner of your Fedora desktop, and it will show a screen with words "Windows Applications". Click on Applications. You will see an icon that says "Install to Disk".
Before clicking on that, ensure that you've got your external drive inserted into the computer's USB port.
Once you click on the "Install To Disk" follow the instructions and get Fedora installed on the external drive.
The drive you are installing Fedora into is possibly identified as /dev/sdb or /dev/sdc. If your drive is identified as /dev/sda, it is referring to your internal drive.