If you are just visiting an ordinary website, nothing dodgy, where there are no forums, etc. is it possible for the website owner to see a record of how many times a particular computer has visited that particular website?
Yep - The web servers generally have logging turned on for site stats and can see which IP address's have visited the site and how many times.
One slight caveat is if your IP address changes each time your connect to your ISP then your identity will change. However their other ways to track you as a person namely cookies and spyware