Free software is software that anybody can download, install, and run for free.
Open source software is software that usually anybody can install, download, and run for free and also program and expand and improve by downloading the source code for that program and modifying it.
The only requirements on Open Source software are that modifications generally be submitted back to the community to allow others to benefit from them.
There are difference sorts of free software as well. Trial versions, demos, and shareware are usually versions of software that are time- or feature-limited, allowing people to understand how the software works prior to committing to purchase it.
Freeware, in it's strictest sense, only applies to software that is unlimited in features or time and is free for all to use. The only limitations on freeware being that the source of the programs are not available for access or modification.
Along a continuum there would be closed-source for install software at one end. Proceeding along the line you'd next encounter trial-, demo-, and share-ware as being "kinda free". Then you'd get freeware, which is free to use but not to modify, and then you'd get open source, which is free to use, free to modify, free to ruin, free to improve... you get the picture.
UPDATE To address modified question:
Here you're getting into poh-TAY-toh versus poh-TAH-toh more than anything else. Though rabid adherents of the FSF or fans of any of the various Open Source licenses would probably claim otherwise.
Yes, as defined, Free Software according to the FSF would not include, in any way, shareware, trialware or demoware. I'm inclined to agree with them. While they are technically free, they are still trying to get you to pay something for more oompah.
Also involved in this ecosystem is "Freemium" software, such as SugarCMS or some of the other enterprise-level applications that tend to be very complex, very robust, and easily comparable to many of their vastly more expensive commercial cousins. These are free to use, but the companies offer enterprise-level support for a cost. Usually a subscription-like fee structure. This allows small outfits or someone with sufficient expertise to run the software for truly free, but companies without the requisite in-house talent can still enjoy high levels of support and availability for a fee comparable to a similar commercial product.
There are many different ideas pulling at GNU, the FSF, and anybody else willing to attempt to codify and corral the herd of cats that are open source and truly free software developers as well as companies building legitimate business models around free software. For instance, Ubuntu is given away free, but the Ubuntu company(corporation?) offers corporate support for a fee, and I believe Red Hat is not offered for free at all (if it is, insert some random non-free linux distro), and some distros are completely and totally free and offer no commercial level support option through an organization or corporation. Are all these open source? Are they all free software? Maybe. Maybe not.
My opinion is that free software and open source represent many of the same things but are not all the same thing. There are signficant points where they overlap, and significant points where they are totally unique.
Some of the points need to be argued over. Companies I work for find themselves unable to consider many open source packages due to the licenses that bind them. Some points ought not be argued over. There are some applications that just aren't worth applying anything more than the most simple and rudimentary of licenses to.