There are two ways to check your mail:

POP (Post Office Protocol) or POP3 in the simplest case just moves your mail from the server to your computer.  Your inbox mail is all on your computer.  You can usually check the box "Leave mail on server" which then makes your e-mail client just copy your mail from the server to your computer.  You can go further and check "Leave mail on server" for "X# of days".  And hopefully this will copy mail from the server, and then delete older stuff from the server so you don't go over quota.

IMAP (Internet Message Access Protocol) or IMAP4 lets you look directly at the server for your email, no moving or copying happens.  Your inbox resides on the server and if you want to read mail there, you need to connect to the Internet to do so.  On some e-mail clients there is an "Offline Folders" option that allows you to copy your mail like POP, just so you can read it on the plane, etc.

Which one to use?

If you have a small quota, then POP is probably the way to go so you don't go over your quota and stop receiving e-mail.  If you have multiple computers, and want to see your inbox from everywhere, it can be tricky, depending on which computer has "Leave mail on server" checked, etc.

If you have a decent-sized quota or are good at organizing your email (ie, putting it into separate local folders), then you might choose IMAP, especially if you work from more than one computer.  This way, your inbox is the same everywhere.  It just takes a little more concentration to make IMAP work if you have a UCD-sized quota.