This can be done, but requires work, hence the popularity of password database software!
You'll want to study mnemonics, the art/science of systems of remembering things. Mnemonic performers do things like memorize the order of a shuffled deck of cards; those methods on a smaller scale should be enough to recall a few passwords.
Derren Brown's "Tricks of the Mind" has a good beginners tutorial, but here's an example of one simple technique, image linking, which should work well for pass phrases.
Suppose your email password is "correct horse battery staple". Imagine, in turn these silly images:
- a huge pile of mail envelopes being corrected by a school teacher as if they were homework
- the same teacher still correcting papers while riding a horse through the classroom
- the same horse in a garage having it's battery changed by a mechanic in oily overalls
- the same car batteries being used to power an enormous stapler
These are deliberately stupid images, the vivider the picture you can make in your mind, the better. Picture this sequence repeatedly, and you'll find that the sequence sticks in your memory; when you start your email program, you'll think of a mail envelope, which leads you to correct, to the horse, then to staple.
Choose words that are easy to visualise; you saw I had a little trouble with "correct" above!
If you don't like typing something that long, use a passphrase initialism, e.g. turn "abicjinh" into "apple bicycle idol chicken joker icicle notebook house" and make image links for those words.
You may have password complexity rules for some apps, where you have to use upper/lower case or numbers. These need a little more work.
Remember which letters are capitalised by picking some landmark from a capital city and setting the image there. e.g. for "aB" you visualise an apple riding a bicycle past the Eiffel tower.
Numbers are harder, the usual way is a "peg" system where you assign a reserved word to each number, memorise those, and then use the associated word in the image. e.g. memorise 1-gun, 2-zoo, 3-tree, then remember "a2" as a wild apple being exhibited in cage in the zoo.