Your designer needs to be well-versed in HTML, which is the coding language for the web. Familiarity with JavaScript is required for things like the mouseover effect you see on many navigation bars. CGI is an acronym for Common Gateway Interface, and is necessary for creating more interactivity on your site, in items like shopping carts, email forms, and surveys. CGI scripts are usually written in the Perl programming language. Paint Shop Pro is another frequently used program.
Illustrator is useful for logo images.
If you plan to use any animation on your site, you will want the designer to be familiar with animation programs. Many simple but effective animations can be done which do not require any special plug-ins, some of which you have seen on this site. Additionally, the designer should make your web site accessible to people with disabilities. This is also important for people using browsers other than Netscape and Internet Explorer. Some of these are text-only browsers.