Active problems are ones that are being worked on by someone or other at a given time. These will be displayed on the main page, so one should feel free to discuss things on the talk page, and other users should feel obliged to help people out with them.
As progress is made on the problem, the article's contents can change - the categorization as "Active" implies that the information is not necessarily correct.
To take on a problem, add to the top of its article:
If you decide to stop actively working on a problem, feel free to remove the active template, and put it into Category:Bountied problems