Orginally, the MDA delivered e-mail to the actual machine that an end user would run their MUA on. So, it was important that MUA's and MDA's agree upon a storage mechanism.Traditionally, this was done using a single file using what's known was Unix mail format. Unix mail format has several short comings, so additional formats were created (the most common of which is mbox). Some MDA's deliver mail to a database rather than files (Oracle Mail is a good example of this).
Then, as personal computers and workstations became more popular, it became likely that people would be running their MUA on a machine that was not necessarily connected to the network all the time, an which was not necessarily setup to act as an MTA. Most importantly people were no longer sharing access to a single machine (to which all mail was delivered). The original solution to this was to have the mail delivered to the end user's home directory, which would then be NFS mounted over the network.
Not surprisingly, the NFS mount solution proved to be a very unsuccessful approach. The performance was terrible, as MUA's frequently had to read in the entire contents of a mailbox before being able to show the latest messages. NFS's file locking problems caused messages to disappear upon delivery. Network traffic went through the roof. Something had to be done. This was when POP (Post Office Protocol) became popular. POP was a simple protocol which allowed a MUA to login to the machine hosting the mail server and download a user's mail messages. To this day most Internet users get their e-mail using this mechanism.