Free software versus open source software
"Free software" and "open source software" are two terms for the same thing: software released under licenses that guarantee a certain specific set of freedoms.
The term "free software" is older, and is reflected in the name of the Free Software Foundation (FSF), an organization founded in 1985 to protect and promote free software. The term "open source" was coined by Christine Peterson and adopted in 1998 by the founders of the Open Source Initiative (OSI).
Copyleft versus permissive
"Copyleft" refers to licenses that allow derivative works but require them to use the same license as the original work. For example, if you write some software and release it under the GNU General Public License (a widely-used copyleft license), and then someone else modifies that software and distributes their modified version, the modified version must be licensed under the GNU GPL too — including any new code written specifically to go into the modified version. Both the original and the new work are Open Source; the copyleft license simply ensures that property is perpetuated to all downstream derivatives.
A "permissive" license is simply a non-copyleft open source license — one that guarantees the freedoms to use, modify, and redistribute, but that permits proprietary derivative works.
There are two kinds of contributor agreements. In a Contributor License Agreement (CLA), the original contributor retains copyright ownership of their contributions, but grants the project a broad set of rights such that the project can incorporate and distribute the contributions as it needs to. In a Copyright Assignment Agreement (CAA), the contributor actually transfers copyright ownership of the contributions to the project, who can then license it however they want since they own it (but a CAA typically grants very broad non-exclusive rights back to the contributor so that they too can use, distribute, sublicense etc their contribution freely).
The following licenses are approved by Open Source Initiative and are popular, widely used, or have strong communities:
- Apache License 2.0
- BSD 3-Clause "New" or "Revised" license
- BSD 2-Clause "Simplified" or "FreeBSD" license
- GNU General Public License (GPL)
- GNU Library or "Lesser" General Public License (LGPL)
- MIT license
- Mozilla Public License 2.0
- Common Development and Distribution License
- Eclipse Public License