Commercial: A commercial license normally means the product must be paid for before use. Products like Microsoft Office, sold in shrink-wrapped boxes, are a good example.
Shareware: Shareware licensing is based on "Try before you buy". Products offered as shareware are normally provided as fully-functional demo copies with an evaluation period. All features will work during the evaluation period, to let you see if the product meets your needs. If you decide it does and purchase a license, you will be given a license key you can apply, which will unlock the product and enable all functions permanently. If you don't get a license within the specified period, some features may be disabled, or the product may stop working entirely.
Freeware: A freeware license means you may use the product at no charge. There are a wide range of variations in this license. In many cases, the product is freeware only for personal use. If you use the product in a business setting, you may be expected to purchase a license.
Some freeware products are "lite" versions of products: the have a set of core functions that will always be available, but have additional functions which can be enabled after purchasing a license. They normally are fully functional as a demo for an evaluation period, and revert to limited freeware form if you don't purchase a license during the evaluation period.
Some freeware products can also be classified as "DonationWare". You are welcome to obtain and use the product free of charge, and it will function with all features enabled. But if you like it and use it regularly, you'll be asked to make a donation to the author as a show of appreciation, to help support further development. A donation is strictly voluntary, and the product will work regardless.
Open Source: An Open Source license means you can get the source code and build, modify, or enhance the product yourself. If you make changes, you are normally expected to contribute your changes back to the maintainer, so they may be incorporated in future releases if desired.
While Open source products are normally also freeware, they don't have to be. There is nothing in the usual open source license that says the author cannot charge for the product, and some open source products do have a cost.
The most common open source license is the GNU Public License, which can be found here: http://www.gnu.org/licenses/
Other common variants include the LGPL (Lesser Gnu Public License), the Mozilla Public License, the Artistic liscense, the Apace license, and the BSD license.
Some open source licenses, such as the GPL, are "viral": there are circumstances where incorporating open source code issued under them in a product you create requires you to issue your product as open source as well.
In all of the above licenses, there will be an author or organization that owns the software, and licenses it to you. If you buy the product, what you purchase is the right to use the product. The author(s) maintain ownership, and the license on the product will specify what you may do with it. In general, you are permitted to use it, but may not re-sell it, disassemble or reverse-engineer it, distribute it without permission, or claim it as your own work.
Public Domain: Products in the public domain have no license. They are explicitly not owned by anyone, and anyone may get them and do what they like with them. This may occur when an author decides he no longer has the time or interest to maintain a program, and will release it to the public domain for the use of anyone who has an interest. The product is freely available, but the author has no responsibility for it, is not required to fix bugs or make enhancements, and is not liable if the product causes a problem for you.