Choose IMAP, not POP.
In Thunderbird, make sure you're choosing IMAP and not POP at the start.
POP: older technology. Messages are sent to the email server you use (Gmail for instance), and stored until you log in with your own PC. When you do, the email is transferred to your PC. It is no longer on the server.
IMAP: Messages are stored on the server as POP. But when you check your email, it simply goes to the server and looks to see what's there. All email stays on server. Can be copied (light bulb icon bottom-left in Thunderbird) to your own PC, which is a good practice to get into for backup.
With IMAP, you can use several computers to check your email, and they all stay in sync, because they all just access mail on the server. So you can check email with the web interface as usual, then use Thunderbird, then go away on holiday and check email, and all will be up to date.