The mailbox password is your way of accessing your encryption key (see What is encryption? for more information). The encryption key is a unique string of characters that allow you to send and receive encrypted messages through Proton Mail. This unique string is generated at the time of account creation and paired to your mailbox password. This pairing turns your mailbox password into the encryption key using the encryption schema AES 256. This allows you to be the sole holder of your encryption key, through your mailbox password.
The mailbox password is not stored on Proton Mail’s servers. Instead an encrypted version of your encryption key is stored on Proton Mail’s server, which is unlocked in your browser with the input of the mailbox password.