Gmail questions…any suggestions?

Why do I see some emails that have my default email address and/or says “On behalf of (my name)”. Yikes. Not sure why some emails show this and others don't. And I just realized that some mailing lists that I respond to will not let me send to it because I'm either not on the mailing list or it's an internal thing, so only company people can email. Does this mean that the emails I send out from my work address through Gmail doesn't act as if I'm using my default Thunderbird account? If that's the case, I'm probably going to have to revert back to my Thunderbird. Any suggestions?