In general, the difference is that open-source code is available to anyone whereas closed-source means the creator of the application keeps the source to himself. For example, the source code of Linux is available at
kernel.org/, but you can't get the source code of Windows anywhere (if you can, it'll be against the law).
If there is something you want to change about a program, then you would typically tell the author about it and maybe he will then make those changes. But with open-source, you also have the option to download the source code, make your changes to it, recompile it and use it straight away. After that you may want to send those changes back to the author. If he thinks it's something useful, then your code could be used in the next official version of the software.
Wikipedia has all the details:
http://en.wikipedia.org/wiki/Open_source
http://en.wikipedia.org/wiki/Proprietary_software
http://en.wikipedia.org/wiki/Comparison_of_open_source_and_closed_source
