If you used IMAP email service with your provider then they should still be on the server. Add your account info to your new install of Thunderbird and check the server.

If you used POP email then the messages were on your computer and are in the recycling center with your old computer.

Please do not use emails as long term storage of wanted items. As you can see that does not work so well. At least make backups of your email.

How do I back up my e-mails? Is that done with you or with a secondary hard drive..or?

Not sure who you think "you" is. Thunderbird is software running on YOUR computer. There is a software called Mozbakup http://mozbackup.jasnapaka.com that you can use to make backups or you can just copy your Thunderbird profile to a safe place.

Are you serious about the "you" comment? You, in the case I used it, is interchangeable with the word "one" or "I" for that matter. "YOU" in the case I used it means just that.

Me, I am a Thunderbird user that volunteers my time to answer questions. I am not going to back up your emails any more than Thunderbird or Mozilla. Thunderbird is email client software. It is not a service. It has no servers to save your mail on. Now is it clear?