The more entropy in a password, the better. Make your passwords long and complex to avoid hackers brute-strength breaking your account.
Do not use the same password on two systems; if one system is compromised, your password on that system can be used to get into your account on the other system.
These two rules mean that if you want to be secure, you need to have a different complex password for each of the systems you use. To make things worse, many of the systems you have accounts on will force you to change your password regularly.
All of the above means that unless you have a photographic memory, you need some secure place, other than your brain, to store passwords.
I recommend using a password manager. They use encryption to secure all of your passwords, and you access them by using one memorized master password that can be complex, since you only need to memorize that one password rather than remembering dozens or hundreds of different changing complex passwords.
The second best option is to print out all of your passwords on paper and store it in a secure place, like your wallet. The real threat to your system accounts is someone hacking into your computer or the system/server you have an account on, not a mugger running home after taking your wallet and hacking into your accounts. Even if you worry about the losing your wallet, you can go home, grab your backup printed password list, and then quickly change all of your passwords before someone gets into your accounts.
One method I recommend that you use to increase the security of any password storage method is to memorize a secret procedure that can modify any alphanumeric string to form your actual password.
For example, I might memorize a procedure that says "capitalize last letter, append 3, and then move first letter to end'. Then if I store my Gmail password in my wallet as 'googlepassword', the actual password I would use when logging into Gmail would be 'ooglepassworD3g'. You would apply the same secret memorized procedure to all of your stored passwords. This allows your passwords to be very complex, since you no longer need to memorize the password, you only need to memorize the secret modification procedure you use.
If you need to change your password afterward, create a new unmodified password, print it out or store it, apply your method, and then use the modified password on that system.
Using this technique, even if you store you unmodified 'passwords' in an insecure location (such as in a text file or in a wallet lost to a mugger), your real passwords will still be secure.
And finally, I believe that passwords will, at some point, go away as a remote authentication method. I have great hopes that SQRL, or some other form of remote authentication, will make passwords obsolete.