What is the difference between POP3 and IMAP4?

All major email clients allow mailboxes to be accessed via either POP3 or IMAP4. These are two different 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 mail, 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). Similarily, 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.

I have received a lot of bounce-back emails for emails I didn't send

This can mean that a spammer has gained access to your mailbox and is using it to send spam emails. There are two main ways this can happen: either the password can be guessed by brute force, or it can be captured by malware on a computer it has been used on. You should change the password to something strong and unique immediately and also run antivirus and malware scans on any computer that has been used to connect to the mailbox.