Email can be accessed 2 ways by Thunderbird.
One way is POP - when you check it that way, the message is taken from the email server and then stored on whichever computer you used to check the mail.
The other is IMAP. With this you are actually looking at the message on the email server. Thus if you look at the message with computer A and then look again with computer B you will see the message and you will see that you have read it.
You can have the messages left in the server if you are using POP as described in the link provided by Bertrum.
If you follow thw instructions there for accessing the account settings, take a look and let us know if it says POP or IMAP
As for your hotmail, you are using the third option for email which is webmail - think of that as a relatives phone number - that doesn't change, but you can call it from any phone - you can use any computer to access webmail.
You can also setup your phone and ipad to get your email - use the imap settings.
If you're not sue how to do that, let us know who you get your email from (you say Plusnet - they're giving you internet, do they give you email as weel?)
We can then find the settings for you.