To keep your emails in sync on multiple computers and/or devices, you should consider using IMAP settings for your mail client.
What is IMAP?
IMAP allows internet users to access their email from any computer or device of their choosing. The biggest attraction is that you can quickly access your email from anywhere with an internet connection, such as the office computer or on the go using your mobile device. IMAP always keeps a copy of your emails on a secure server which offers great redundancy for backups and is the reason why the your emails stay in sync.
What's the alternative?
The alternative to IMAP is POP which used to be the preference before the internet became so readily accessible. However POP acts like a post box - once the mail has been delivered, that's the only copy that you'll get. If something should happen to the system / device where you've received that mail on, unfortunately all of the mails would be lost.
Can I change from POP to IMAP?
Your email account can be configured on either POP or IMAP so changing between the two is straight forward. All that is required is for the email account to be setup on the system / device of your choosing (and the old account removed) and you can then carry on like normal.
See the 'Email Setup' section for all email account setup guides to get the change started.