No, there is no charge for downloading and using Skype, if you want to talk to another person on their computer that also has Skype on it.
However, if you want to call someone who does not have Skype, or maybe not even a computer, then you do have to buy credits from Skype to make a call, which you do from your own computer to call their telephone. You can use either a headset and mic. or, as I do, a webcam (mic. built in) and speakers. You can also get a dedicated Skype phone handset that plugs into a USB port. Spoilt for choice really!
Good luck.