In most people's minds, a "professional" design is relatively straightforward. What do you imagine when you think of a businessman? Probably a cleanshaven, wellgroomed man in a nice white dress shirt, dark colored jacket/blazer, and dark slacks, with polished shoes, right? Well, even if your image may differ, it surely isn't a homeless man wearing a hawaiian shirt and bright blue and green swimming trunks, is it?
Just like that, when people say "professional", they want to see a color scheme that's usually mono or dichromatic (1 color w/ variations of the shade or 2 distinct colors + variations). They also want everything neatly layed out and they shouldn't have to go hunting for links to other pages. Many hosting sites, for example, have flash banner headers, but these too must be professional. They do not have giant pink scrolling text with glitter and stars floating around... it simply has a picture of a server or a technician slowly transitioning between other images.
Just think of it this way: you are asked to present your website to... Donald Trump (or insert some other rich businessman). Would you show him something crisp, clean, and "professional" or something flashy, colorful, and abstract?
Hopefully that somewhat cleared it up, but the concept in general is hard to convey with words.