Are you talking about Thunderbird on the Vista computer?

I have a new windows 10 Computer that I want to bring mail from my old Vista computer to.

Both, Vista and Win10 are operating systems. You don't read mail with an operating system but with an email client (or a browser). So you'd need to explain what data from what application you want to migrate.

Both operating systems include a mail utility, I want to transfer from the one that comes with Vista to the one that comes with Windows 10, which doesn't include an import feature. I'm hoping to use Thunderbird on Windows 10 in place of the included mail function. I want to be sure Thunderbird will allow me to import things the way they are organized in the mail utility included with Vista. I hope this is finally making sense.

You still haven't given us the name of the mail app on Vista. You must bear in mind that some of us have never had to use Vista.

I take it you don't use IMAP and so your messages are locked into your computer and not on a server, when it wouldn't matter which client you use. To check, go to your email provider's website and look for your email messages there. If you can see them then you don't have a problem. Just install this account into Thunderbird and you'll have all your messages on your new computer.

If that doesn't work out for you then you may need to move messages. To import into Thunderbird, it needs to coexist with the donor email app. So a possible first step would be to install Thunderbird on the Vista machine and see if it can import from the incumbent mail app. If this works, you can move the newly created Thunderbird profile from the old Vista computer to the new one.

However, it's probably easiest to create a new IMAP-connected account (e.g. gmx, gmail), set it up on the Vista machine and in Thunderbird on your new machine, and start copying messages into it in the Vista machine. These will appear via the magic of IMAP synchronization in Thunderbird on your new computer where you can move them back into your preferred accounts.

And then there are your address books...

All I can find about the name of the mail app is Microsoft Mail. The messages are on my computer, no longer on the mail server. I will try adding Tunderbird to the Vista machine. Will Thunderbird "break" anything already existing? Once I have Tunderbird installed, what do I do to copy my messages into it? Thanks!

I believe that Windows Mail used the same file format as Windows Live Mail which replaced it: https://en.wikipedia.org/wiki/Windows_Mail Both programs stored individual messages as .eml files and these can be imported directly into Thunderbird. Here is a thread describing the process by someone who has done it before you: https://support.mozilla.org/en-US/questions/1046033 As for contacts data, I believe they can be exported as a single .csv file: that's certainly the case with WLM. That file can be imported into the Thunderbird address book: https://support.mozilla.org/en-US/questions/1055905