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Free Software is software that respects users' freedom and community. The term was coined by Richard Stallman.
Roughly, "free software", means that the users have the freedom to run, copy, distribute, study, change and improve the software. Thus, “free software” is a matter of liberty, not price.
Free Software is often clarified as "free as in speech" rather than "free as in beer". It's also called, “libre software”, borrowing the Latin-originated word for “free” as in freedom, to show that it doesn't mean, "gratis".
Origin of Free Software goes back to the foundation of the GNU Project. As Stallman was already within the hacker culture of MIT and shared the source code of every utility he made under the GNU project, he decided to make his own movement based on that culture. 
Free Software is consisted of four freedoms, which are:
- Freedom 0: The freedom to run the program for any purpose.
- Freedom 1: The freedom to study how the program works, and change it to make it do what you wish.
- Freedom 2: The freedom to redistribute and make copies so you can help your neighbour.
- Freedom 3: The freedom to improve the program, and release your improvements (and modified versions in general) to the public, so that the whole community benefits.
Those freedoms are mostly protected with Free Software Licenses, that divides into two as permissive and copyleft ones.
Free Software Licenses like GPL and CCASL protect four freedoms with copyleft, which is a type of copyright license that pursues those freedoms as the only obligatory term, so people can't simply steal your software to be shared as proprietary. 
Permissive Free Software licenses such as BSD, MIT and MPL include but don't enforce the four freedoms for derivatives of the original source code, but makes it obligatory to include a copyright determiner to the original source code, which makes it defenseless to being used in proprietary software, which is a difference as compared to copyleft licenses. As they are weak at the protection of software freedom, Free Software communities generally don't prefer permissive licenses, as the freedom isn't preserved for derivatives of the software.
The difference in origin, motivation, and ideological viewpoint for the term "Open Source" in contrast to "Free Software"
Generally the media refer to Free Software as Open Source, or use the term of FOSS (Free Open Source Software) to avoid confusion with the word "gratis software". Free Software and Open Source, however, are two separate movements that have different backgrounds and goals. Free Software is the name of the movement that supports four software freedoms, while Open Source, as a market strategy from its origin, just supports the source code to be open for rapid development of new companies that need free labor.
As a consequence, "Open Source" tends to be a term favored more by corporations, whereas free software and associated terminology are the original terminology of the movement that both of them refer.
False Claims about Free Software
There are common false claims that are made about Free Software in various platforms.
Definitions of "OSS" and "FOSS" and "Open Source being an umbrella term"
It is said by OSI and its adherents, that "Free Open Source Software" determines software which are free (in speech or in gratis) and "Open Source Software" defines "all so called market-friendly open source software that you can make money". That is completely false, as the term Free Software exist long before since "Open Source" is invented by Eric S. Raymond.
Confusion of Permissive Free Software being Open Source
Some people say that software that have the permissive licenses such as BSD and MIT are Open Source licenses as they let enterprises to use their work and open the code without protecting the four freedoms. That is completely bullshit that Open Source Initiative and groups that support it say, as in reality there are no so called Open Source licenses that aren't Free Software as well.
Odd licenses that are just "Open Source" but not Free Software
There are some odd licenses accepted by Open Source Initiative as "Open Source" that aren't even a such according to their manifest. The most famous one of those is the NOSA (Nasa Open Source Agreement) license, that is only usable by NASA for their inner projects, which is simply a trade secret that is labeled with the word "open". Those licenses are literally referred as a counter-argument to the fact that there is no Open Source license that isn't Free Software.
Rhetoric of "You can't make money with Free Software!"
That is a famous rhetoric mostly said by open source advocates, that are actually paid by big enterprises to do that job as Open Source Initiative has donations given by Microsoft and other big companies. You can actually make profit with Free Software and with the Copyleft licenses to be specific, as companies like Redhat, Ngninx, Apache and Mozilla are one of the most successful ones. 
The Unicorn Law
Basically, women in free software programming are still considered unusual enough that they're tokenized, and then asked to talk about being a token minority as if all that matters is their gender. This applies to pretty much any industry where programming, computing, or other technology is involved.
Odd Free Software Movements
Nationalist Free "Open Source" Software Groups
Some nonsense that is quite popular among racists and conspiracy theorists, especially within the gray wolf movement, that suggests Free Software can be used for liberating a nation from "Judeo-American International Enterprises and Secret State" rather than the user and programmer. They totally miss the point of what Free Software actually stands for and generally steal free software to make their national rip-offs. Most of those people generally don't know how to code or design, as well as important terminology but have bare knowledge about editing a code and replacing the assets. The most popular examples of nationalist free software are:
- Pardus, a Debian clone maintained by Turkish Government, that has controversial names at its versions
- Red Star OS, a (yet again) Debian clone maintained by North Korean Government, specifically designed to look like MacOS as Kim Jong Un once seen using Mac on a photo and denied it by saying it is their national operating system.
- İlbilge, a "free" search engine.
Religious Free Software
There are also several religious projects that are generally ripoffs of existing free software. Most notable examples are:
- Christian Ubuntu, that doesn't let the user open non-christian content.
- Helal Linux, an Islamic GNU/Linux distro based on Ubuntu, that is the same with Christian Ubuntu.
- Xiphos, a Bible study tool.
"Open Source" Religions
That is actually a thing. That is a name given to religions that have copyleft-like views regarding property and use the 4 freedoms to be spread within the internet, yet identify themselves within the odd market term "open source". There are several religions that use that methodology, such as:
- Discordianism, a religion found in 70s, even before both Free Software movement and the term "Open Source".
- Open-source Judaism, an odd movement started by a jew that followed GNU movement during 80s.
- Open-source Yoga, a religion that supports public domain, that isn't neither Free Software, Copyleft or "Open Source".
- Open-source Wicca, a religion born from a book made by a guy.
- Dudeism, a religion that is declared as "open source" by its founder.
- Richard Stallman, the guy behind of all
- Linus Torvalds, a guy in the middle
- Open source, the market term
- Eric Raymond, racist that started a market term and made money out of it
- GNU website
- Frequently asked questions about GNU licenses
- Official Website of Free Software Foundation, a good source of information
- Open Source Initiative's Website, a good example of bullshit
- OYAKDER, the hotspot for Turkish Free Software nationalists
- History of The GNU Project
- GNU Manifesto
- What is Copyleft?
- List of the entire GPL-Friendly Free, Non-GPL-Friendly Free and Non-Free Software Licenses
- Why Open Source misses the point of Free Software?
- bs website called opensource.com
- Makeuseof's bs article about the topic
- Microsoft makes it official
- Free Software Companies
- Geek Feminism article on the Unicorn Law
- Article of BusinessInsider