People that say their computer runs Linux usually refer to the kernel, tools, and suite of applications that come bundled together in what is referred to as a distribution.
Take Linux and the GNU tools, add some user-facing applications like a web browser and an email client, and you have a full Linux system. Individuals and even companies started bundling all this software into distributions almost as soon as Linux became usable. The distribution includes tools that take care of setting up the storage, installing the kernel, and installing the rest of the software. The full-featured distributions also include tools to manage the system and a package manager to help you add and remove software after the installation is complete.
Like UNIX, there are distributions suited to every imaginable purpose. There are distributions that focus on running servers, desktops, or even industry-specific tools such as electronics design or statistical computing. The major players in the market can be traced back to either Red Hat, Debian, or Slackware. The most visible difference between Red Hat and Debian derivatives is the package manager though there are other differences in everything from file locations to political philosophies.