We use a technology called "cookies", which are widely used across the Internet. Cookies are small text files that are created by a web server and stored on your computer when you visit a website. They don't do anything i.e. they are not executable code and can only be read by (you and) the website that created them. You can view and edit the cookies on your computer like any other text file using a text editor (the contents are usually just strings of unique identifiers and date/timestamps). The website that created the cookie can read the contents when you are at their website.
All the major browsers allow you to block cookies and delete those that have already been created on your computer, usually within the 'Tools' section of the browser. These tools allow you to specify which cookies you will accept by type and often by specific websites using an exception list e.g. you can block all cookies and then list the website from which you will accept cookies.