The IP address of every visitor can be recorded in the server logs. They can also record the URL of the page that you were at before you arrived at their site.
If you access the site from a fixed IP address it may be possible to trace the owner. This is fairly easily done by a reverse DNS lookup if your company hosts web services such as email or a web site. They can't get your personal email address but they could get enough of the address of your website or email administrator to contact them.
Most (not all) domestic users have a dynamic IP address that changes each time they reconnect to the ISP. Unless the IP address has a DNS entry nobody would know who is there except your ISP.
However the server log can be matched with the ISP logs to trace phone line used to access a site. This is how police sometimes catch cyber criminals.
bubbles - when you click on a website there is no way for the website to immediately know who you are or to know your email address unless you explicitly give it that information. The best the website can do is to know roughly where you are, possibly to the nearest big city, but it can get that very wrong. There are ways to try to trace you, but that requires the cooperation of various organisations and even then there are no guarantees they will be able to find you.
we have a second home in the usa. to cut a long story short an ex from way back who used to knock me around has tried to contact me under another name and using a mobile number. i traced his number to a business carpentry website and i wondered if he could see that i had been on there looking. i do not want him to know where i am as you can imagine!