There are two big organizations: FSF (Free Software Foundation) and OSI (Open Source Initiative), and while they have similar ideas, they disagree on the details. Free software is more in line with the ideas of the FSF (more idealistic), while open source is more in line with the idea of the OSI (more pragmatic).

There is also the term "libre" (meaning "free as in freedom") to distinguish it from software that doesn't cost money.