A password is only as strong as it is random. You cannot test for randomness of a password invented by a user in his brain, because randomness is a property of the generation process, not of the result. Your Web site will see only the result.
However, you can exclude passwords which are too short because short passwords are amenable to a very primitive exhaustive search, which means trying all combinations of letters up to a certain length. The main risk you must avoid is to antagonize users: passwords will be only as strong as the user chooses so, therefore you must make him a voluntary contributor to the site policy. The user, by himself, is not very interested in the site security. He enters the password because he has to (the site enforces a password entry), but he sees the password as a hindrance: he did not came to the site to enter a password, but to access whatever pages the site contains.
This boils down to the following recommendations:
Enforce a minimal password length of 8 characters. Not much more than that; but people are accustomed to entering 8-character passwords. If you ask for more, users will protest (and they protest by choosing "witty" passwords, which are not strong at all).
You may require a mixture of letters and digits. But don't overdo it. Letters and digits are fine; don't enforce uppercase/lowercase mix or punctuation: these are noticeably harder to type on smartphones and tablets.
Provide an optional password generator. Since human minds are not good at generating randomness, use the computer. But be careful not to make the generator mandatory: the user must see it as an helpful tool, not an unavoidable regulation. Therefore, provide a button "generate a good password" which generates a sequence of lowercase letters and digits (not an overly long one, mind you; 8 characters).
Publish guidelines which explain what a good password should be (i.e. random). Users love to feel empowered. Also, warn them about usability issues with regards to non-ASCII letters: accents, non-latin characters... will make their life much harder if they use them.