Your ISP or internet service provider does keep a log of the websites you visit and your activities while on the internet through their services.
So if you do something illegal on the internet and the courts summon those logs then the ISP will look them up and provide the logs and details of the websites you visit to the court.
Usually though the ISP is not looking at no caring about what websites you visit or what you're downloading because they would not have enough time nor manpower to keep a check on what websites everyone is viewing or what they're downloading.
So while the ISP can tell what websites you're visiting or what you're downloading they're not even looking at it unless required to do so for a court order.