What is the name of the file that my emails are stored in on my computer?

There are many. So perhaps you might want to share why you are asking. This is a topic on which hours if not days could be spent without much trouble.

Thanks for responding. When I open Thunderbird, nothing appears in my inbox. I have to manually download my messages and normally that takes several tries. Once I get the messages down loaded I want to be able to copy or save them permanently on my computer so I don't lose them.

Okay, I seriously mean no offense, but your statements are so open that we could discuss at length, as Matt indicated. :) So, seriously, tell us your intent. For example, one solution to your issue is to do daily backups. Or maybe you just want to save a few important messages. In truth, once you see a message, it is already on your PC, but that doesn't necessarily mean it's where you can find it. Thunderbird is terrific, but it wasn't designed to do permanent stuff. Again, for example, I use a backup service (free, there are many) that runs daily and copies my selected folders to the cloud. Free. Daily. I also do a full backup PRIOR to any thunderbird upgrade. Why? If there are problems, I can easily restore. Rarely needed, but practical. I also do a full hard drive backup twice a month. Hey, you never know. :) So, tell us your intent, and we will offer what we can. And you're very welcome here. :)