'Skype' is a company which offers a 'VOIP' - that's 'voice over internet protocol' - service. 'Skype' is also sometimes used generically, in much the same way as 'hoover' or 'poratakabin' are, to refer to all VOIP services.
Basically, instead of your phone being a tin can at one end of the telephone company's long piece of string, with the phone of the person you're calling being the other tin can, yours and their computers are the tin cans, and the whole of the internet is the string. Because you are already connected to (and paying for) the internet, the call costs nothing.
With Skype, you install a bit of software on your computer, and you register an account (for free) under a unique username, which acts much the same as a unique telephone number. You can then make calls to anyone else with a Skype account who is online at the same time as you, for free. You can make voice calls, or video calls (if you have a webcam).
You can also use the Skype system (and other VOIP systems) to access the regular telephony system - ie you can call real telephones, anywhere in the world using your computer. This costs money, because you are accessing telephone company string, but it is (in general) cheaper than just using your mobile or landline. To do this you buy credit from Skype.
Finally, you can also purchase a 'telephone number' from Skype if you want, which can be useful for a variety of reasons. People using mobiles or landlines will be able to dial that number to reach you, just like a regular telephone number.
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