What did I do to violate the Google Webmaster Guidelines and get kicked out of Google?

Yow. That is a scary thing to find once you log in to your admin account!

The good news is that if you aren't trying to "game" the search engine, then it's likely that you've inadvertently done something that's easy to fix and yes, you can get back into their index if you request a reinclusion.

What you must do immediately is go and study the Google Webmaster Guidelines, take notes on anything you think might be a mistake you've added on your pages, and then methodically ensure that each and every page of your site is clean of any problems.

Common problems include having foreground and background colors too similar, or identical (white on white is a classic search engine spamming technique, and easily identified), having different pages for search engine spiders than users (called "cloaking"), pages that contain identical content to other pages on your site or other pages elsewhere on the web (called "duplicate content"), and loading up pages with irrelevant words ("keyword stuffing"). Fundamentally, the best advice I'll give you is exactly the same as what Google suggests:

That's really the key to all of this, and if more people focused on creating the best possible content on their site, unique, fresh content that met a genuine need in their user community or market segment, we'd have less "blackhat" seo and more great results from a given search.

Okay, off my soapbox. :-)

Once you believe that you've cleaned up everything that might be a problem on your site, you can request that the Google folk reevaluate your site, check for whatever problems where previously there, and, hopefully, get back into the index. Start here:
Submit a reinclusion request.

Can it be done? Definitely. I know of a number of sites that have been banned for violations of the Google webmaster guidelines and, upon cleaning up their act, have shown back up in the index with good SERPs and all. What percentage are reapproved? I have no idea, but I bet it's not that high...

Good luck. Please do come back in a week or three and let us know how it went!