Is Postfix then my client ?
No, Postfix it the Mail Transport Agent, the glue between servers to transport mail. I assume you use a setup like:
Personal PC --> network --> Server
On your server, you run programs like Postfix (to transport mail) and POP3 (to connect the client and the server to retrieve mail). As a substitute for POP3 you can use IMAP. Both serve the same objective, though they use different protocols. For Postfix there are also a couple of other MTA's regularly used, like sendmail. On your PC client( often a windows PC, but other flavors become more and more popular), you use a mail client like Outlook, Outlook Express or Thunderbird to name a few.
To sent mail you use the server directly over the so called SMTP protocol. Thus your Personal PC client software (Outlook, etc) connects over the network with the SMTP protocol to the MTA (Postfix) on the server.
So it's POP3/IMAP to retrieve mail, SMTP to send mail. For POP3/IMAP often a package is used from a team of developers under the name Courier, but there a lots of other options. Perhaps the overwhelming amount of options is a bit scary, but try to read as much as you can. Running a mail server is do-able, but will require a significant learning curve. Try google for more information on setting up a mailserver, mail client usage or similar. I'm pretty sure useful information will popup.