Free software is software that gives users the right to use the software, to modify the software, and to pass on the software, modified or not, all free of charge and without restrictions on what the software is used for.
The term free software was used already in the late seventies, but was codified by Richard Stallman through the definition of the four software freedoms in 1986. The definition and publication was performed through the Free Software Foundation, a U.S.-based non-profit organization founded by Stallman in 1985.
Open-source software provides users with the same rights as free software. For all practical purposes, they are the same.
The term open-source software was defined by the Open Source Initiative, a U.S.-based non-profit organization founded in 1998 by Bruce Perens and Eric Raymond. The definition was written by Perens and consists of a list of ten criteria, which Perens derived from the Debian Free Software guidelines.
Open-source software is often abbreviated as OSS and free and open-source software is often abbreviated as FOSS. Sometimes the term libre is thrown in to emphasize the freedom to do what you want and to downplay that the software is free of charge. This leads to free/libre, and open-source software, abbreviated as FLOSS by non-native speakers of English.
Not everyone agrees that free and open-source software are the same. Free software proponents argue that users who receive free software must also be given access to its source code, even if modified by the provider of the software. To enforce the right of a user to receive the source code, Stallman invented the copyleft obligation, which requires that anyone who distributes free software cannot change the license terms. This prevents software vendors from keeping proprietary modifications of free software locked up.
Rights to and obligations for free and open-source software are codified as free and open-source software licenses. The copyleft obligation became popular as part of the GNU General Public License 2.0.
In contrast to free software, open source proponents rely on enlightened self-interest of software vendors to contribute their modifications to open-source software projects. They don’t try to force anyone who distributes open-source software to lay open any modifications if they don’t want to. Open source licenses which grant the rights listed above but don’t contain a copyleft obligation are called permissive licenses, in contrast to copyleft licenses, which contain such a clause. Common examples are the MIT license, the Apache 2.0 license, and the BSD family of licenses.
For the better part of the nineties and the noughties, the philosophical debate about software freedom and copyleft allowed the enemies of free and open-source software to spread fear, uncertainty, and doubt about the usefulness of open-source software and arguably delayed the dominance we see today by a decade or two.
Today, free and open-source software are part of almost all existing software, proprietary or not.