To understand what the public domain is, you first need to know a little bit about copyrights. The copyright system has changed quite a bit over the years, and seems to be more focused on corporate interests than on public interests. Basically, when you create something you automatically gain copyrights on the item that you created. You can not copyright ideas, just the particular implementation of an idea. Copyrights last a set period of time. The particular amount of time varies by country, with the US being the life of the creator plus 70 years but corporation owned copyrights last only 95 years from publication or 120 years from creation (whichever is shorter) due to the fact that corporations are legal entities that can theoretically live forever. For the duration of the copyright, the owner controls what can be done with the work, though there are some exceptions for educational use.
When the copyright term expires, the material enters something that is known as the Public Domain. This simply means that anyone who wants to can take the public domain material and do whatever they want with it. To be perfectly blunt, the vast majority of material that has reached public domain status is too old to be of any real use. this is one reason a lot of people want shorter copyright terms. I personally would like to see 10 year copyrights that can be perpetually renewed for another 10 years for a small fee and the renewal information would be publicly accessible so it would be easy to see if something was still under copyright. The advantage of such a system is that anybody would be able to keep their copyrights forever if they desired, but less valuable material would fall into the public domain sooner so that there would be a much greater chance that it would be useful to society.
While the vast majority of the public domain is so old that it is not relevant to modern society, there is still a lot of material that can be used. For instance, Shakespeare has stood up to the test of time quite well. While you can still pay for a printed copy of a Shakespeare play, many archive sites have free copies of the plays that you can download. If you want to put on a play, you don't have to pay any type of public performance fee for using the material. In fact, you can even modify the play however you like. And that is where the problem is.