Passwords are (if the implementation is good) stored as a hash code in your system. Furthermore they should get salted to hide weak passwords (in case someone gets a grip of the database). You can read about salts and password storage here: http://en.wikipedia.org/wiki/Salt_%28cryptography%29
If you read the article you will understand, that done right the password itself will never get stored. What gets stored is the hash code of the password + salt and the salt itself. What you could do to try your system for weak passwords is the same thing that hackers do: using brute force. In your special case you could (if one salt is used for all passwords) use a table with passwords + salt and the generated hash code. This will decrease the computation time rapidly, as you will only have to compare the hash codes (This only is true if you use the database more than once). But again, this is only possible if the implementation is not the best possible solution.
If have not forbidden weak passwords and you want your users to use strong passwords the easiest (only) way to achieve this is by forcing the users at password generation, or checking their password at login as long as the password is still stored clear in memory. Thus you can only easily check for strong passwords for users "using" the computer. If you want to check the passwords of all users your option is brute force.
If you have not done it in the past the solution would be to reset ALL passwords with random generated strong passwords and hand those to your users. At the next login you can force your users to use strong passwords.
John the Ripper is a brute force attack. It has a massive dictionary and stored hash codes and then runs this against your passwords. You could always run that, but should be a waste of CPU time, as you should enforce strong passwords.