A brainwallet refers to the concept of storing Bitcoins in one's own mind by memorizing a seed phrase. If the seed is not recorded anywhere, the Bitcoins can be thought of as being held only in the mind of the owner. If a brainwallet is forgotten or the person dies or is permanently incapacitated, the Bitcoins are lost forever. Using memory techniques allow them to be memorized and recalled easily.
Brainwallets are not recommended to be used in general because of fallible human memory. But in special situations they could be very useful, for example when fleeing a country as a refugee with only the clothes on your back.
- On a computer with no malware, run Electrum and generate a seed phrase.
- Memorize the phrase using http://en.wikipedia.org/wiki/Mnemonic_peg_system
- When spending or saving, restore the wallet from memory using the phrase.
- Use the master public key to create an online watch-only wallet, where you can send to but not spend.
- Spend from the wallet in the manner of deep cold storage. Transferring the unsigned transaction to the cold storage computer, signing it and broadcasting to the network.
Example Mnemonic Peg
To memorize a seed with this method you must invent a story which hits the words as "keynotes". Try to make it like a fairy tale story, use imagery. Make it somehow striking and emotionally resonant. When remembering you just remember the key words, not all the other words - the other can be remembered more as images and thoughts (which are hard to write down)
Let's say we have this seed:
witch collapse practice feed shame open despair creek road again ice least
You'd imagine walking through a building familiar to you, maybe your own home or workplace or school.
- You imagine looking in the first room and seeing your mother dressed as a witch, playing the jenga boardgame until the tower collapses.
- You walk to the next room and see your father practising with a longbow, he shoots a chicken to feeds himself.
- In the next room you see your brother naked in shame attempting to cover himself, he's looking through a window that's open and flapping in the wind.
- Now you reach the kitchen, girlfriend is looking at Picasso's Guernica on the wall. She is in despair from it. Next to it is a television playing the show Dawson's Creek.
- Next you're in the garage, your childhood friend is working on his car. He plans to go on a road trip for the 5th time this month, he's going again.
- Finally to go outside to the garden. It's early spring and the ground is covered in melting ice. Two of your other friends are there, one friend has a huge basket of apples, the other has a smaller basket but you're holding only some apples. You've got the least apples.
Repeat this story in your head several times over a short period - the first few days. It will sink in, deep, after that. You'll only have to revisit it very occasionally. After a while you can ignore it for months and it'll still come back, not that I'd recommend relying on that.
Video Example of Mnemonic Peg Method
From the BBC documentary The Human Mind (2003) by Professor Robert Winston. Approximately 31 minutes in. Memorizing a list of 30 random words.
Fallible Memory Warning
Despite the memory aids, human memory can be very fallible. So if your only storage is memory you may find that it just vanished one day.
Data should always be backed up. Storing a seed phrase in one place is bad, even if that one place is your brain.
Obsolete Brainwallet Style
An early old-style brainwallet was created by by memorization of a passphrase and converting it a private key with a hashing or key derivation algorithm (example: SHA256). That private key is then used to compute a Bitcoin address. This method was found to be very insecure and should not be used. Humans are not a good source of entropy. Using a single address also has problems associated with address reuse.
Low Entropy from Human-Generated Passphrases
Practically everyone who knows about or cares loudly yells at people DO NOT USE BRAINWALLETS [GENERATED BY HUMANS]. We've seen pretty concrete evidence that users are resistant to good advice in this space, and they are shocked when their favorite quotation is cracked and they lose their coins (But it was 60 characters long! I even added a special character! how is this possible?!), the existing sites promoting this stuff won't use a KDF stronger than SHA256*1 because "users are stupid if they use weak passwords". 
Ryan Castellucci DEFCON Talk
Ryan Castellucci gave a talk at DEFCON23 about cracking brainwallet passphrases. Although brainwallet passphrases were being exploited for years by this point, the talk helped bring the issues to more popular consciousness.
If you have coins in an old-style brainwallet, the website http://www.bitaddress.org/ contains a GUI for generating the private key using the sha256(passphrase) algorithm. It's highly recommended you move the coins out as soon as you can.