Hi and welcome to LQ!
Yes. Linux is free and open source. Some exceptions are the Enterprise editions, like Red Hat (RHEL) and SuSE (SLES) for which the charge is due to the offered services (customers support, automatic updates and so on).
You can choose one Linux distribution, download the DVD iso image, burn it and reboot your system from the DVD to start the installation process. Many distributions offer a LiveCD version to test the system without actually installing it on your hard drive.
You can also have a look at LQ ISO
, a section of LinuxQuestions where you can find links to download major and minor distributions. You can find lists of the most popular distributions and some reviews.