What is a VPN?

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It sounds pretty cool, hearing you can browse or surf the net in private without anyone seeing what you do. Why I ask is because I have a cousin who I think snoops in on what I do on my laptop, whenever I get on to use his router he gets on his computer as well, when I get off so does he, when I get on he does too. Thing is I don't like, I would confront him about it but I want to try this vpn out first.

So my question.
1.What is a VPN, specifically. (Describe in your own words first please before/if you send me a link)
2.Is the VPN only for someone who has a router? Like say I decide to get internet at my place from whatever company I choose, would a VPN only work for the router I buy? Or can it still work if I get it (VPN) for my laptop?

Please answer as best as you can and feel free to throw in any additional details I should know about, thank you!