Passwords are in the news thanks to Gawker media, who had their database of userids, emails and passwords hacked and published on the web. A big part of the fault is Gawker's, who was saving user passwords (so it could email them) and thus was vulnerable. As I have written before, you should be very critical of any site that is able to email you your password if you forget it.
Some of the advice in the wake of this to users has been to not use the same password on multiple sites, and that's not at all practical in today's world. I have passwords for many hundreds of sites. Most of them are like gawker -- accounts I was forced to create just to leave a comment on a message board. I use the same password for these "junk accounts." It's just not a big issue if somebody is able to leave a comment on a blog with my name, since my name was never verified in the first place. A different password for each site just isn't something people can manage. There are password managers that try to solve this, creating different passwords for each site and remembering them, but these systems often have problems when roaming from computer to computer, or trying out new web browsers, or when sites change their login pages.
The long term solution is not passwords at all, it's digital signature (though that has all the problems listed above) and it's not to even have logins at all, but instead use authenticated actions so we are neither creating accounts to do simple actions nor using a federated identity monopoly (like Facebook Connect). This is better than OpenID too.