These days, all major email clients (Microsoft Outlook etc) allow mailboxes to be accessed via either POP3 or IMAP4
POP3 and IMAP4 are what we call "email protocols".
On your desktop client, POP3 is downloaded directly from the mail server onto your computer, your computer will store a "local copy" of the email and usually delete it from the server. You can create local folders on your computer to organise the mail, but you won't see these or your mail if you logged in using webmail. Additionally, if you create folders to organise your mail in webmail, you won't be able to access these on your desktop client.
Mail will therefore only contribute towards the mailbox limits while it is left undownloaded in the inbox.
IMAP4, in contrast, is left on the server (where it will continue contributing towards your mailbox limit). If you create folders/organise your mail in your desktop client, you will see this update when you log in with the webmail client (and vice-versa). Similarly, when you mark a message as "read" in webmail, it will appear "read" in your desktop client.
If you're accessing your mail on several different computers, IMAP can, therefore, be useful.
What is the best protocol to use?
If you are an "on the move" kind of person who wants to access your email on a desktop computer as well as a smartphone then your best option is to use IMAP4.