I'm assuming you already understand something about passwords, like the fact that "12345" an anything in the English dictionary are terrible choices for a password. I don't delve much further into cryptography or security theory, because it would take too long, it would be distracting, and mostly because I'm not an expert in that field (hopefully you will see that as a plus, since otherwise I would spend the whole time scaring you).
The basic problem with passwords is that if you have enough strong passwords to have adequate security for all your different web sites and other situations, then it is difficult to remember them all. Solutions like writing them down or storing them in a spreadsheet solve those problems while introducing a host of other risks. In addition to the obvious security holes, keeping that information up to date can be a bother.
Now that I have a family, I realized there are other needs too. I handle most of the family finances, but if I am hit by a bus, whether I survive or not, how will my wife figure out all the stuff needed to take over? Less important but more common, if my wife gets a new frequent flier number and password, is there a good way to share it so that either of us can access it when needed?
If my laptop is stolen, or my house is burglarized, am I going to have to change all of my passwords? If you are starting to catch my drift, you can see that the first step is to think about your risks. While most people are affected by the same risks: burglary, malware, injury, stupid mistakes, natural disasters, some situations may be more important to you than others, so it's still worthwhile to give some thought to your specific concerns.
Most of this is not terribly new; many people already use a password database stored on the cloud, but I haven't seen much written on how to share it securely. Even if you don't need to share your passwords with a family, you will probably still find some or all of this system to be useful.