Same tip can be used to start multiple instances of other programs.
The only way to run 2 instances of Skype presently is:
On a Windows OS you can run another Skype instance at the same time if you start the second with another Windows user account. Example
1. Make sure you have another user account (login) for your Windows.
2. Right click the Skype icon and select "Run as..."
3. On the next screen click "The following user:" and select one different from the one you are currently logged in as on your computer.
4. Voila - another Skype opening which can be used independently from your first Skype instance. You can even make calls between them on the same computer
Additional tip from JP White.
To help automate the 'run as' trick you can create a second desktop icon for Skype and modify its properties. Click the advanced button and check the box 'run with different credentials'. After clicking on the modified icon it will ask you to sign in as another user. Changing the icon graphic also helps ensures you click on the correct icon.