Skype works fine behind a firewall or router or whole bunch of them. But you must know what your firewalls are blocking and provide an unblocked port for Skype to use.
First make sure you install the latest version of Skype; there are frequent updates.
Then go through all your Skype configuration settings. The one you MUST pay careful attention to is this one:
Tools - Options - Connection - "Use Port xxxxx for incoming connections"
Skype will suggest a randomly-selected port. But you can specify any port you want to use. (It does NOT have to be the same as the port used by anyone you call.)
The trick is to specify a port your firewall is not blocking! This port is used to improve the sound quality of your Skype conversations.
For security reasons, there is no one standard port used by Skype. I suggest you use one of the obscure, higher numbered ports, but I will NOT recommend one to you! You can choose a port up to 65536 ! If you want to learn more about common port assignments, do a Google search. Gibson's website may also provide some useful information:
But you really do NOT have to do this research. Just pick a port number and make sure you don't block it in your firewall settings!
You can also check the box right after: "Use port 80 and 443 as alternatives for incoming connections". If you do, you obviously do not want your firewalls to block these ports.
Then choose, "Automatic proxy detection", unless you have a special reason to select one of the other choices and know what you are doing.
I hope this helps you and clears up the mystery. Skype is really very easy to use. And it's fun too!
Survivor @ Ground.Zero