Validation is a process of checking your documents against a formal Standard, such as those published by the World Wide Web Consortium (W3C) for HTML and XML-derived Web document types (XHTML), or by the WapForum for WML, etc. It serves a similar purpose to spell checking and proofreading for grammar and syntax, but is much more precise and reliable than any of those processes because it is dealing with precisely-specified machine languages, not with human natural language.
It is important to note that validation has a very precise meaning. Unfortunately the issue is confused by the fact that some products falsely claim to "validate", when in fact applying an arbitrary selection of tests that are not derived from any standard. Such tools may be genuinely useful, but should be used alongside true validation, not in place of it.
In simple terms, validation ensures that your website complies with the standards accepted by most web designers. This allows your site to be accessible to more people, across more web browsers, operation systems, and search engines.
First, there is the very practical issue that non-valid pages are (by definition) relying on the error-correction of a web browser. This error correction can and does vary radically across different browsers and versions. Many authors who unwittingly relied on the quirks of Internet Explorer 6.0 suddenly found their pages appeared totally different in Internet Explorer 7.0. While Internet Explorer initially set out to be bug-compatible with Netscape, it too is moving towards standards compliance in later releases. Other browsers differ further. And we haven't even discussed the differences between the same browser running on different operating systems.
The novice (or non-technical website owner)
The answer to this one is that markup languages are no more than data formats. So a website doesn't look like anything at all! It only takes on a visual appearance when it is presented by your browser.
In practice, different browsers can and do display the same page very differently. This is deliberate, and doesn't imply any kind of browser bug. A term sometimes used for this is WYSINWOG - What You See Is Not What Others Get (unless by chance). It is one of the principal strengths of the web, that (for example) a visually impaired user can select very large print or text-to-speech without a publisher having to go to the trouble and expense of preparing a separate edition.
It is perhaps unfortunate that the best-known browsers - FireFox and Internet Explorer on Windows - are visually very similar in their presentation of many documents, differing only in trivial details like margins and spacings. The "same" browser on a Mac or Unix/Linux display will often look far more different.
The perceptive observation
REMEMBER: Household-name companies expect people to visit because of the name and in spite of dreadful websites. Can you afford that luxury?
Even if you can, do you want to risk being on the wrong side of a lawsuit if your site proves inaccessible to - for instance - a disabled person who cannot use a "conventional" browser? Accessibility is the law in many countries. While validation doesn't guarantee accessibility, it should be an important component of exercising "due diligence". It is now almost 8 years since a court first awarded damages to a blind user against the owners of a website he found inaccessible (Maguire vs SOCOG, August 2000).
The strawman argument
This is simply head-in-the-sand ignorance (it lies at the heart of the most spectacular hype-filled dot-com failures). Validation is fully compatible with a wide range of dynamic pages, multimedia presentations, scripting and active content, etc. It is part of the difference between doing it right and doing it wrong in a dynamic multimedia presentation, just as much as in a purely textual site.
It is perfectly OK for authors to express their creativity on the Web. It is generally more appropriate to some sites (e.g. recreational ones) than to others (e.g. informational or functional sites). But authors with creative ambitions should bear in mind that in any artistic field, you must start with a thorough understanding of the rules before breaking them. Otherwise you just look foolish.