Once you've booted into the Linux installer, it's just a matter of stepping through prompts.
The Fedora installer, Anaconda, presents you a "menu" of all the things you can customize prior to installation. Most are set to sensible defaults and probably require no interaction from you, but others are marked with alert symbols to indicate that your configurations can't safely be guessed and so need to be set. These include the location of the hard drive you want the OS installed onto and the user name you want to use for your account. Until you resolve these issues, you can't proceed with the installation.
For the hard drive location, you must know which drive you want to erase and re-image with your Linux distribution of choice. This might be an obvious choice on a laptop that has only one drive, to begin with:
Screen to select the installation drive
Select the drive to install the OS to (there's only one drive in this example)
If you've got more than one drive in your computer, and you only want Linux on one of them, or else you want to treat both drives as one, then you must help the installer understand your goal. It's easiest to assign just one drive to Linux, letting the installer perform automatic partitioning and formatting, but there are plenty of other options for advanced users.
Your computer must have at least one user, so create a user account for yourself. Once that's done, you can click the Done button at last and install Linux.
Anaconda options complete and ready for installation
Anaconda options are complete and you're ready to install
Other installers can be even simpler, believe it or not, so your experience may differ from the images in this article. No matter what, the install process is one of the easiest operating system installations available outside of getting something pre-installed for you, so don't let the idea of installing an OS intimidate you. This is your computer. You can and should install an OS in which you have ownership.
Own your computer
Ultimately, Linux is your OS. It's an operating system developed by people from all over the world, with one interest at heart: Create a computing culture of participation, mutual ownership, and co-operative stewardship. If you're interested in getting to know open source better, then take the step to know one of its shining examples and install Linux.