Can someone's router I'm using see what I do on my laptop?

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So say I go to a friend or relatives house and use their wifi for internet. Since they pay for it and the modem or router they use belongs to them, is it possible they can gain access to whatever I do on my own laptop or see what I do? I had to ask this because every time I use my cousin's router he always gets on his computer, whenever I get off he does, when I get on so does he. If he can see everything I do I don't feel comfortable with it, and don't tell me to confront him about it. If he can see what I do I want to beat him at his own game. Is there a way I can keep what I do private so nobody else sees? Or can private surfing only work for routers? I'm confused about private networking between computer/laptops and modems/routers. Please explain to me, would it be illegal?