You are familiar with passwords, your secret login credentials. Account Keys are longer, full 'phrases' that add security while often being easier to remember.
Because an Account Key is longer, you don't need to worry about adding numbers, symbols, and mixed-case letters to make it strong. A few random lower-case words will be relatively easy to remember, while substantially increasing your account's security.
When you sign up, Peerio will automatically generate an Account Key for you.
Peerio uses eight-word Account Keys to achieve an estimated 120-bits of entropy, which is estimated to cost over $1B (US) to crack in a year. Peerio also uses scrypt to make cracking even more costly in both time and money.
It is important to remember your Account Key, write it down on a sheet of paper or print it out from Peerio on your computer and store it securely. If you lose your Account Key, you lose your account!
If you are still a bit confused about why we use passphrases, this comic from xkcd explains the issue excellently: