I use a configuration like this:
set mbox_type=mbox
set folder=~/Mail
set copy
set spoolfile="/var/spool/mail/myusername"
Normally, you'll fetch the mail with fetchmail. Fetchmail speaks protocols like IMAP and POP which are used to GET your mail from somewhere else. Fetchmail than usally delivers the mail to a MDA - a mail delivery agent. I use for example qmail for that. Qmail knows where the mailboxes in my system reside and puts all mail into a mailbox. Where exactly the mailbox is and wether the mailbox is an "mbox" (all mails appended into one big file) or a "maildir" (a directory with one mail = one file) is up to you and your configuration.
I let qmail spool my mail into /var/spool/mail/myusername - this is my mailbox. There it lies until I open mutt. Mutt checks the spool-file in /var/spool/mail/myusername and shows the usal mail listing.
To complicate things, I use procmail to filter mails. That means: instead of letting qmail spool all incoming mail into my standard spool file in /var/spool/mail/myusername, qmail delivers the mail to procmail, procmail does all kinds of filtering - including throwing away annoying stuff - and sorts my mails in to several folders ALL residing in ~/Mail/ - mailing lists for example get their own mailfolder. Everything NOT matching specific procmail rules ends up in my spool file in /var/spool/mail/myusername.
You'll need four things: fetchmail to fetch mail, an MDA to deliver and transfer mail around (use what you like) in your system, procmail for filtering, possibly also spamassasin for filtering spam specifically and your MUA (mail user agent) to actually read and write mail - mutt in our case. As qmail and/or postfix also can act as MDA, this distinction seems to look a bit nitpicking, but there're also tools for Linux/Unix which actually do delivering only - not sending out, no acting as a "real" mail server - which qmail and postfix both can.
And we're not talking OUTGOING mail here - incoming only.