There are many different e-mail clients available. One popular freeware client is Thunderbird, produced by Mozilla, the authors of the popular Firefox web browser.
The most common e-mail client is probably Microsoft's Outlook. Outlook allows you to create a backup file of all your e-mail messages and contacts. This backup file is in a .pst format, and can be used to move your files from one computer (or e-mail client) to another.
When switching to Thunderbird, you may experience difficulty in importing a standalone .pst file, since Thunderbird uses .mbox files instead and does not recognise .pst files. Workarounds do exist to import a .pst file.
Use Outlook to decode the .pst file. If possible, install Outlook on the computer with Thunderbird and the rogue .pst file. Once Outlook is installed, select "File" > "Import and Export" to import the .pst file into Outlook. Once the messages/contacts have been successfully imported, open Thunderbird and select "Tools" > "Import" > "Mail within Thunderbird." Choose "Outlook" as the default client from which to import.
Install Thunderbird onto another computer. If you can't install Outlook onto the desired computer (referred to here as "Computer 1"), install Thunderbird onto a machine with a compatible version of Outlook (referred to here as "Computer 2").
Open Outlook on Computer 2. Copy the .pst file onto Computer 2. With Outlook open, select "File" > "Import and Export." In the window that pops up, choose "Import from another program or file" and click "Next."
Choose "Personal Folder File (.pst)" and click "Next." Use the "Browse" button to navigate to the location of the .pst file and click "Open." Everything in the backup file--e-mails, contacts, etc.--will be imported into Computer 2's Outlook.
Import the files on Computer 2's Outlook into Computer 2's Thunderbird. Open Thunderbird and select "Tools" > "Import." Click the "Mail" button and click "Next."
Click on "Outlook" and click "Next." When the import is finished, click "Finish."
Close Thunderbird. Click "Start" and navigate to "Your-username\AppData\Roaming\Thunderbird\Profiles.default" (where the is a string of random characters). Copy all of the included files and folders to a flash drive and move the flash drive to Computer 1.
Navigate to "Your-username\AppData\Roaming\Thunderbird\Profiles__.default" on Computer 1. Copy the files and folders from Computer 2 into that directory. All of the email info will now be available on Computer 1's Thunderbird.
Try using the Thunderbird PST Import plug-in (see Resources). This is an open-source plug-in that purports to make the Thunderbird client recognise and import .pst files. It is not clear whether or not it works; many of the comments on the download site state that it is both unclear how to use it and that it does not work as advertised.
Always back up your data. When importing files, there is always the possibility that the files may become corrupted or illegible by the computer.