Assume you set up pop3 and imap and want to check it out.

First, let’s go to check the POP3 server. Make sure that it bound at port 110:

# netstat -lnp | grep 110
tcp        0      0 0.0.0.0:110                 0.0.0.0:*                   LISTEN      2779/dovecot

As you can see I use dovecot as server application and it’s works on all interfaces.
Next, make connection attempt:

# telnet 127.0.0.1 110
Trying 127.0.0.1...
Connected to localhost.localdomain (127.0.0.1).
Escape character is '^]'.
+OK Dovecot ready.

Now, we got server answer. Do authorization with command user and pass

USER user1
+OK
PASS qwe
+OK Logged in.

and LIST to see headers of all inbox mails:

LIST
+OK 1 messages:
1 306
.

We have only one mail. First field of answer means

mail_id

.
To see whole mail use RETR mail_id:

RETR 1
+OK 306 octets
Return-Path: 
Received: from d (node1 [10.0.30.1])
        by mail.test.com (8.13.8/8.13.8) with SMTP id n8ONtVLG011279
        for user1@test.com; Fri, 25 Sep 2009 03:04:08 +0300
Date: Fri, 25 Sep 2009 03:04:08 +0300
From: fd@test.com
Message-Id: <200909250004.n8ONtVLG011279@mail.test.com>
test

To delete it:

DELE 1
+OK Marked to be deleted.

This message marked as deleted and will be removed when you send command QUIT to server. To unmark messages use command RSET without any arguments:

RSET
+OK

Use command QUIT to leave the server.

QUIT
+OK Logging out.

Refer to RFC 1939 for more details.

# netstat -lnp | grep 143
tcp        0      0 0.0.0.0:143                 0.0.0.0:*                   LISTEN      2779/dovecot

Try to connect:

# telnet localhost 143
Trying 127.0.0.1...
Connected to localhost.localdomain (127.0.0.1).
Escape character is '^]'.
* OK Dovecot ready.

Okay, we have got good answer. Now, we should authorize on server. Syntax is:
? LOGIN USER/ALIAS PASSWORD

For example:

? login user1 qwe
? OK Logged in.

To get folders list make:

? LIST "" "*"
* LIST (\NoInferiors \UnMarked) "/" "INBOX"
? OK List completed.

In folder INBOX contain all incoming messages. To select it:

? SELECT INBOX
* FLAGS (\Answered \Flagged \Deleted \Seen \Draft)
* OK [PERMANENTFLAGS (\Answered \Flagged \Deleted \Seen \Draft \*)] Flags permitted.
* 1 EXISTS
* 0 RECENT
* OK [UNSEEN 1] First unseen.
* OK [UIDVALIDITY 1253837819] UIDs valid
* OK [UIDNEXT 2] Predicted next UID
? OK [READ-WRITE] Select completed.

1 EXISTS – means 1 incoming message. You can choose between see full message or only message body instead:
? FETCH message_number All
OR
? FETCH message_number Body

Let’s see whole message:

? FETCH 1 all
* 1 FETCH (FLAGS () INTERNALDATE "25-Sep-2009 03:04:12 +0300" RFC822.SIZE 306 ENVELOPE ("Fri, 25 Sep 2009 03:04:08 +0300" NIL ((NIL NIL "fd" "test.com")) ((NIL NIL "fd" "test.com")) ((NIL NIL "fd" "test.com")) NIL NIL NIL NIL "<200909250004.n8ONtVLG011279@mail.test.com>"))
? OK Fetch completed.

Exit user command is ? LOGOUT.

To learn more about IMAP refer to RFC 3501.