It acts as a proxy. So instead of your connection going PC -> Website and your ISP recording that, it goes PC -> VPN and the ISP can't see further than that. Usually all encrypted so can't be snooped either. Obviously should they wish the VPN provider could keep a list of what you access, so you have to pick wisely, but there are good reviews out there. The good ones don't keep logs of what anyone visits so even if the police turn up asking for your porn history, they can't give it to them.
Deep/dark web is something different. It actually refers to any part of the web not crawled by search engine, so company internal sites, webcams, etc. But also something called Tor, which is what the media mean when they say dark web. It's a collection of websites that use a special protocol called Tor that requires a certain browser to decode. Then the entire Tor net sort of acts like a VPN in that you don't connect direct to a deep web site (or onion link as it's known) you connect to other Tor users, so again your ISP/Government/Russian Hacker Wife only sees you connecting to another user, not the encrypted content you receive via them. Issue with Tor is that if you are an "exit node" you are on the edge of the network connecting to dodgy sites for people and you know what sites people are going to or can man in the middle them. So it was in the law enforcement's interests to put themselves into the network as exit nodes so they could spot what people were doing. There's some debate about how secure Tor is now.
