A website should have a unified graphic theme, as well as a tone and general feel. If a "professional feel" is what you're going for—if you're trying to sell a professional product or service, like web design—then aim for a professional look and design, and write the content with a professional tone. It's hard to describe because the concepts are often abstract, but a good designer can come up with a look and feel for any theme.
Besides helping the user subconciously identify your website with a general category (professional, playful, serious, etc.), it's important to have a unified theme so that the user knows when they're actually at your website. That is, from a cognitive psychological view: if a user clicks a link, he should know whether he is still at your website or if the link has taken him to an external site.
Distinct and unique "logo types" (amazon.com, ebay, Microsoft, RollingStone Magazine) stay with a user. They are instantly recognizable and also lend an appropriate air to the content they represent.
It can take a long time to develop and decide upon a theme, but the visual aspect of a website is just as important as the backend and is an essential part of the user's experience.