Assuming that you already have a Skype session session running:

Linux

In Linux, press “Alt + F2” to open a launcher. Type: This will launch a second instance of the Skype client where you can login to another Skype account.

Windows

In Windows, press “Windows key + R” to launch the Run dialog. Type: If you are running the 64-bit version of Skype, use this command instead:

Mac

In Mac, there is no simple workaround to launch multiple instances of Skype client. The easiest way is to use the app – Multi Skype Launcher to open a second instance of Skype.

Download the Multi Skype Launcher app, extract it and click on it. It will prompt you to enter your password. Once that is done, you will see a second Skype client show up in your Mac. You can then login to the secondary account.

Conclusion

For both Windows and Linux, it is best that you create a desktop shortcut with the command so you can launch it easily. For Mac, you just have to place the “Multi Skype Launcher” in a handy position. Enjoy!