Answer: "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. Like the FSF, the OSI's founders supported the development and distribution of free software, but they disagreed with the FSF about how to promote it, believing that software freedom was primarily a practical issue rather than an ideological one (see for example the entry "How is `open source' related to `free software'?" from the OSI's original 1998 FAQ page).
Many who later adopted the term "open source" broadly shared the ideological perspective of the FSF but had some disagreements over strategy and rhetoric. Today some people use both terms, choosing according to context and audience.
One of the tactical concerns often cited by adopters of the term "open source" was the ambiguity of the English word "free", which can refer either to freedom or to mere monetary price; this ambiguity was also given by the OSI founders as a reason to prefer the new term (see "What Does `free' Mean, Anyway?", and similar language on the marketing for hackers page, both from the original 1998 web site).
In the 1990s, the term "open" applied to software source code was sometimes used to imply source code being merely inspectable or visible or available. Going back further, in the 1980s there were uses of "open" in the computing industry that primarily connoted something like "absence of hardware vendor lockin". OSI's term "open source", as defined in the Open Source Definition, makes clear that open source specifically entails not mere inspection access but also conveying to recipients the perpetual right to fork covered code and use it without additional fees.
The FSF uses a shorter, four-point definition of software freedom when evaluating licenses, while the OSI uses a longer, ten-point definition. The two definitions lead to the same result in practice, but use superficially different language to get there.
This history has led to occasional confusion about the relationship between the two terms. Sometimes people mistakenly assume that users of the term "open source" do not intend to communicate a philosophical point of view via that term, even though many actually do use it that way. Another mistake, which has occasionally been seen since about 2008, is to assume that "free software" refers only to software licensed under copyleft licenses, since that is how the FSF typically releases software, while "open source" refers to software released under so-called permissive (i.e., non-copyleft) licenses. In fact, both terms refer to software released under both kinds of license.
Neither term binds exclusively to one set of associations or another, however; it is always question of context and intended audience. When you sense a potential misunderstanding, you may wish to reassure your audience that the terms are essentially interchangeable, except when being used specifically to discuss the history or connotations of the terminological difference itself. Some people also prefer to use the term "free and open source software" (or FOSS, FLOSS [free, libre and open source software]) for this reason.