Humans are not usually capable of remembering too many different passwords. You either use a small set (often one) of passwords, or you use one 'master' password that accesses a password vault (as described above by Phantom). Either way, you're not using lots of different passwords.
The advantage of the password vault is that the password shouldn't leave your computer, but it's not impossible. It also means that you've got all your eggs in one basket and if something happens to that basket (stolen, breaks down unexpectedly, or gets swept away in the great coffee flood of '14) then you've just lost access to all your accounts. If you back up the passwords in the vault, then it's no longer secure, and you've got the problem that there's multiple places from where the passwords could be stolen.
I know people who keep their set of passwords small. They'll have a unique password for each critical system, and a common password for non-critical accounts. That's a compromise position between the same password for everything and a password vault, with a known risk for the accounts that the person doesn't consider to be critical.
There's no perfect solution. There are, however, some solutions that are better than others (and solutions which are really poor...)