If you want messages deleted from the server when they're deleted from TB, most users would prefer an IMAP setup (gmail recommends 'just mark as deleted' for IMAP). Most POP users would set 'delete from server after download' and then messages deleted from TB would go to the local Trash folder.

I don't see this option "delete from the server after download". What I'd like to achieve is to keep certain emails on my PC and to have only those email in Gmail. For now when I delete emails from TB same emails are not deleted from gmail. Is there an easy way to do this?

For POP, the actual setting is 'Leave messages on server', which, if unchecked, means 'delete from server after download'. If you keep it checked, but also check 'Until I delete them', I'm not sure deleting in TB will delete from the server. It might happen only when the Trash is compacted.

https://support.mozilla.org/en-US/questions/1128358#answer-890862

As stated before, the easiest way to sync TB with the server is with IMAP.

The link explanation that you sent works form my ATT accounts, but not for gmail. SO far I didn't find a way to keep only the messages I want both on my PC and on the gmail. Are there instruction somewhere to change POP3 to IMAP? Do I have to remove this account and add it again? Where can I find the exact settings for both TB and Gmail?

http://kb.mozillazine.org/Convert_a_POP_account_to_a_IMAP_account

The setup wizard in TB should find the correct gmail settings: imap.gmail.com, 993, or pop.gmail.com, 995, smtp.gmail.com, 465; SSL/TLS, OAuth2, email address.

http://kb.mozillazine.org/Gmail

I switched from POP3 to IMAP and looks like deletes are wotking now. Thanks for your help!