I will attempt the challenging, and seemingly controversial, task of recommending a few Linux distributions for the new Linux user. A few things before we start:
Regardless of what you may have heard Linux has come a long way in the past couple decades. The vast majority of Linux distributions provide easy to use installers that do not require any Linux specific knowledge to pull off a basic install. If you can succesfully accomplish a new windows install you can probably do this as well. In fact if you have an interest in trying Linux at all I think it is safe to assume that you have, at least, some basic computer knowledge which should suffice.
Also take some time to try different distros out. This is the beauty of Linux; it is totally free (Yes, it will cost you some bandwidth to download the images). Try any distribution that strikes your fancy and find out which one works for you. If you have the capability VirtualBox or VMware works well for this. In a virtual machine you can try out multiple distros with out a bare metal install or inadvertantly crippling your only machine. You can even launch several distros and try them out side by side.
Another option is most distros provide live dvd images that allow you to try before permanently installing the OS. If none of these are an option, and you only have a single machine, try to have an internet capable device handy such as a smartphone. If you run into trouble Google is your friend. There is a vast Linux community out there and if you encounter a problem it is guaranteed that someone else has run into it before and found a solution as well.
Well lets get started!
It is with great trepidation but I have to go out on a limb and say that this is my number one pick for users new to Linux looking for a good Windows replacement. It is packaged with a "hand holding" installer that can not really be any simpler. The default Cinnamon desktop environment is very attractive, intuitive, and should have a familiar layout for users coming from a Windows environment. In my experience hardware drivers are probed and installed without any coaxing. The only exception to this may be proprietary video card drivers. This will sometimes require an extra step after installation to take advantage of hardware acceleration but is usually a very simple process. Mint is an offshoot of Ubuntu with the Mint team focusing on creating a beautiful Linux desktop experience. You will have access to Ubuntu's expansive software repositories and easy to use package manager.
Debian is also a good option for the beginner. Debian is in the group of some of the most mature Linux distributions still being maintained. Many other popular distributions, such as Ubuntu and all of it's derivatives, are actually based on the stable Debian project. Installation is still fairly simple although when choosing an installation image you are presented with a great deal of choices which may be confusing if you don't know the significance of what you are choosing. A solid distribution choice but not as polished (for the desktop user) as some of the others.
Fedora is an exceptional desktop distribution and in some ways superior to Ubuntu derivatives like Mint. Installation is also very easy although maybe slightly less "hand holding" compared to the previous example. The Fedora team, an offshoot of and sponsored by Red Hat, has a focus on upstreaming cutting edge open source code. This means a very fast update interval and the most current, stable Linux kernel, which is awesome, but also that closed source and proprietary software and drivers take a back seat. This may mean some extra steps or difficulty finding and installing those proprietary hardware drivers, although that has never been my experience. Fedora also has great software repositories and an easy to use package manager. The Gnome desktop is a bit odd but if you can not get used to it Fedora comes packaged with several other alternative desktop environments. This is the case with most distributions so if there is a desktop that you know you do not like check for alternative desktop environments prepackaged with that distro you want to try.
CentOS is not a distribution that you will often find on a beginner list as it is one of the distros often associated with and, depending on who you ask, synonymous with enterprise Linux. It is another derivation of RHEL (Redhat Enterprise Linux) but focuses on being rock solid, stable. You will not find bleeding edge kernels or software here (unless you install them manually) but you will find a complete linux tool chain and an OS that takes pride in it's stability. I found it to be simple to install and use. The default desktop environment, though not heavily polished, is also intuitive and easy to use. And there is good software repositories as well as a powerful package manager. A solid ditribution choice for any user.
A very lightweight derivitave of Ubuntu. As with all of my recommendations installation is simple and hardware drivers should be taken care of for you except, of course, in the case of proprietary drivers and problem hardware. The Bohdi team's focus is on creating a very lean distribution that does not come prepackaged with a lot of extra software. Enlightenment, the default desktop environment, is also attractive and easy to use while being very light. It will run well on very antiquated hardware so a good choice if you want to breathe new life into that old laptop or desktop.
Just like CentOS, openSUSE is a distribution that is, unfairly, not often considered for beginner Linux users. There are two major install options, tumbleweed and leap. Tumbleweed is a rolling update model that upstreams cutting edge software and may suffer from stability issues. Leap uses a more conservative update model and may offer increased stability. The installer offers more than basic options but if one chooses the defaults it is every bit as simple to install as the rest of these distros. However, as a disclaimer, if you are installing in a virtual environment be prepared for difficulties with graphics acceleration. You may want to install with a desktop environment (There will be an installer prompt), enlightenment or xfce come to mind, that does not require hardware acceleration. If you are installing on bare metal Nvidia gpus are supported well but even these may be an issue with Tumbleweed. If these issues do not scare you away this is a solid distribution with room to grow. It may have a steeper learning curve than some of these other distros but that is half the fun, right?