Freeware

Closed source, copyrighted proprietary software, which costs nothing to download and install.

Open Source

Software which the authors have published including all the source code needed to rebuild the project. This software either includes it's own source code with the application, or includes an easy way to obtain the source code (such as a url embedded in the application and/or it's documentation).

The two major categories of open source software:

Copy - Center

Closable source software such as software using the MIT license or BSD license. Code of this type can be added to most any other type of software, including closed source commercial software. Public Domain open source software also falls under the closable source category, the main difference being that projects incorporating public domain open source code DO NOT have to include any copyright notice from or license reference to the original authors.

Copy - Left

Non-Closable source software such as software using the GPL license (GNU General Public License). When adding GPL licensed code, all your modifications, additions to the code, and generally your entire project must be made freely avalible to the public. Additionally, the GPL verion 3 assigns any patents you may have registered and used with a product using any GPL v3 code to the public, so anyone and any other company may use your patents
free of charge. For these reasons, some people consider this license type to be "viral".

Commercial Software

This software type generally requires full payment up front before it can be used. Most software purchased in boxes at stores is this type. Commercial software is also bought and sold for digital delivery on the internet, usually at a slightly lower price than the boxed version.

Shareware
Try before you buy software. Shareware can be freely downloaded, but usually has time or feature limits before it has been fully paid for. A program set to stop working after 30 days is the norm. Also, some shareware includes "nag screens," a window which pops up and reminds the user the program is not free, often upon loading the software or using certain features.

Beyond these license categories:

The future will define major new licenses and license categories fostered by technological innovation and changes in the law.