What is freeware? — Freeware is software created by professional developers or hobbyists, and distributed for free (over the internet, typically, or bundled in other software packages). Most of the time, freeware is developed by a single programmer. In some cases, you will find programs that are a "hybrid" form of shareware and freeware, where a feature-limited version of the application is completely free, and a more functional version of it is sold as "shareware" (software you have to pay for). Often, you can try non-free software for 30 to 60 days, to let you try-before-you-buy, but this is not freeware - just a "free trial". Here's an example - download a free trial of Outlook 2007: it will work like the regular version, but you'll have to purchase a license key (aka "serial code") after two months, if you want to keep using it (that same kind of deal often comes with new PC's, with shareware pre-installed). Pictured left are the logos of the Firefox web browser and Thunderbird email client - some of the most well know freeware programs in their category (which happen to also be "open source - concept explained at the bottom!)
Side info: the term freeware is generally used for downloadable software that needs to be installed on your computer. Many online programs (or "web apps") are free as well, but generally not called "freeware" - there is no logical explanation for this, it is just usage.
Here are a few freeware email programs: all of them are fully functional - you never have to pay for them after a certain time, or to unlock some features. Except for the last, all are "cross-platform" desktop applications: no-one needs to stay away from email for financial reasons!
* These two programs are "open source", not just freeware: explained in the next section.
Note: email programs like Outlook Express (XP), Windows Mail (Vista), Windows 8 Mail (Windows 8), or Apple Mail (Mac OS X), come bundled with their respective operating system: in that sense, they are a core component of Windows or Mac OS X - and not "freeware" per se.
Unlike regular freeware, Open Source has a philosophical component: open source developers want software to be free for all, and create programs whose "source code" is open to anyone to see and modify. The source code is the set of computer instructions that programmers type inside specialized text editors (often "IDE") to make the software we all use. These developers collaborate online, and very often come from completely different backgrounds, countries, and languages (both their mother tongue and computer programming language of choice!)
The Open Source movement and its volunteers are responsible for widely popular pieces of software like Mozilla's Firefox web browser and Thunderbird email program, the FileZilla FTP software to upload files, and operating systems like UNIX (the foundation of Mac OS X, which itself is "proprietary software", not open source), and Linux (with very popular "distributions" like Ubuntu and Mint).
Here's the official website of the Open Source Initiative: OpenSource.org