At the risk of sounding like an inexperienced idiot, here goes:

I have two computers and one cable modem and multi-port router.

I've never considered connecting both computers to the router at the same
time.

Here's my question: If I connect both computers at the same time - and if
both are turned on, how will my incoming e-mail be routed? Will all
e-mails go to the inboxes of both computers? If not, what happens?

Or, am I better off connecting only one computer at a time to the router?
Or, is it ok to connect both {live} at the same time?

Thanks - james