If I was at a hotspot in say Singapore, would an email client like Thunderbird retrieve my ISP email from my Australian ISP's mail server via the net? Is this how a mail client actually works?
Possibly - and sort of!
Your incoming messages are stored by your ISP on a mail server before being downloaded to the local message store in the email client (internet mail is a "store and forward" system) on your PC.
The traditional protocols for retrieving mail are POP3 and IMAP - your local email client connects to the server directly to log in and download messages, and to send messages.
Note that older POP3 servers will not store messages after you have connected to the server and downloaded them, whereas an IMAP server gives you the option to leave copies on the server.
This goes back to the days when dial-up was the norm, and calls were costly, so messages were sent / received as quickly as possible, then the call disconnected, and mail could be read and composed "off-line".
The problem is that - for security - you generally cannot directly access that server from the wider internet, but only from "inside" your ISP's network - which you are with your broadband connection from home. Some ISPs may allow pop3 / imap internet access to their email servers, but mine certainly doesn't!
Having said that, if you have a dial-up modem connection, you would be coming in on the "inside" of the ISP network, and you could dial-up from anywhere in the world (at a cost), so you could access email directly.
More recently, it has become possible to access the message store on the mail server at your ISP via a web browser - but ONLY if your ISP provides that service, obviously. Of course you then have the problem of differences between your local message store and the server message store.
I think that this form of web browser email access can sometimes be used by a local client such as Tbird or Outlook Express to connect and actually download messages to the local mail store, but again not all ISPs' email systems will support this. Your best bet is to contact your ISP to see what mechanisms they support.
SD