Do you still have access to the other computer?
If so, you should export it in a workable format. If the other computer had Thunderbird, the default format selected is (comma separated (system encoding .csv)), but there is also comma separated (UTF-8; csv); tab separated and vcard. The default format is the one that works the best for such transfers. If the other computer is using Outlook or some other email, you could export in CSV or some other tabular format it offers. If you have more than one choice, do many exports and try to see which one works the best.
Unfortunately, there might be incompatibilities in formatting between what your previous email exports and what Thunderbird wants, especially if you use a language other than US English.
LibreOffice is a great free program that allows you to import CSV files and transform their encoding to and from UTF-8 (for accented letters and apostrophes), field separation, and field naming (i.e. what you have in the first line of the file).
Finally, if you transfer from another program (or even from Gmail Contacts), some fields have a slightly different name. So try to import, and if some of the fields don't have what you would expect them to have, open your CSV file (again) with LibreOffice and change the headers accordingly.