You can record and view a list of web-sites you visit on your home router, but that feature is usually buried and not turned on by default. Some routers might not even have that feature. Some manufacturers make you register your router to access this information. Other people are correct in that anyone with the administrator access to your router and disable certain web-sites.
There are a couple of things you can do to prevent people from looking at the log files assuming they exist.
1. Change the router administrator password. You should do this anyway because most routers use something easy to guess like "admin" or "password."
2. If you cannot change the administrator password, then periodically delete that history in your router.
Private browsing only prevents a log file from appearing on the PC you're browsing from; the router would have a record of it (assuming that the feature is enabled)
ISP's do keep logs of all internet activity, but it's a lot of information to sift through. Further, there would have to be a good reason to release that information like for a police investigation. I don't think they can do anything for a suspicious spouse.