When you buy a household electrical appliance, you can choose from a range of options. You don't need to ask "does the power supply in my house support Hoover or Electrolux"; you know it supports both, and you are free to choose. That's because there is a standard for mains power, that is observed by everyone concerned - the manufacturers, the power supplier, and the electrician who wired your house.
Exactly the same argument applies to the Web, unless someone subverts it. The difference is that fewer end-users have the confidence to make a fuss when they've been sold a dud. Valid markup is one of the standards that make it work.
 If those aren't household-names in your country, substitute other brands as appropriate.
Validation is a process of checking your documents against a formal Standard, such as those published by the Worldwide Web Consortium (W3C) for HTML and XML-derived Web document types, or by the WapForum for WML, etc. It serves a similar purpose to spellchecking 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 nebulously-defined 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", whilst 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.
Well, firstly there is the very practical issue that non-valid pages are (by definition) relying on error-correction by a browser. This error correction can and does vary radically across different browsers and versions, so that many authors who unwittingly relied on the quirks of Netscape 1.1 suddenly found their pages appeared totally blank in Netscape 2.0. Whilst Internet Explorer initially set out to be bug-compatible with Netscape, it too has moved towards standards compliance in later releases. Other browsers differ further.
There are also three specific questions we should deal with here:
Click on any of the question marks to see our answer here.
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 coincidence). It is indeed 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 - Netscape Navigator and MS Internet Explorer on Windows - are visually very similar indeed 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.
Do 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. Whilst validation doesn't guarantee accessibility (there is no substitute for common sense), it should be an important component of exercising "due diligence". It is now just over a year since a court first awarded damages to a blind user against the owners of a website he found inaccessible (Maguire vs SOCOG, August 2000).
This is simply head-in-the-sand ignorance (indeed, 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 in order for authors to express their creativity on the Web, though it is of course generally more appropriate to some sites (e.g. recreational ones) than to others (e.g. informational or functional sites like this one). 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.