- Viewing any errors the search robot may have crawling your site
- Setting your domain's geographic target
- Setting your preferred crawl rate (frequency that Googlebot visits your site)
- Setting your preferred www or non-www domain format (not necessary if you already use 301 redirects in an .htaccess file)
- Finding duplicate title and meta description tags (so you can make sure each one is unique)
- Viewing your top keyword referrals
- Enhancing your 404 page to include what Google thinks the visitor was looking for and a Google search box
- Setting which site links you want to appear in search results (example below):
But that's not all, you can also check out which of your site's pages have links from other sites (the more relevant links you have from respectable websites, the better). This can help you recognize links from bloggers (maybe it's positive or negative feedback), partners and other sites. It can also help you track your link building / marketing strategy -- just don't expect the tool to show you every single link Google knows about.
You can also use the "page not found" report to locate URLs that other sites have linked to that don't exist on your page (a typo or a page that doesn't exist anymore). You can "save the link" by redirecting that link to the proper page, or contacting the webmaster of the other site to fix the link.
There are cases when you need to block search engines from crawling certain pages or areas of your site. For example, your staging site and secure pages. Other times you might find a rogue URL that slips into the Google Index that you don't want in there, like a URL with a session ID. The worst case is someone Google's your website and that session ID appears instead of your home page, with the page title "ERROR" -- I've seen it happen.
Google's Matt Cutts describes how to prevent search engines from indexing your pages, and how to use the URL removal tool if you need to in Google Webmaster Tools:
Get slapped with a Google penalty? You can use Google Webmaster Tools to request re-inclusion (after you fix the reason why you were penalized).
Google's not alone in offering webmaster tools. Yahoo and MSN (Bing) offer their own too. You'll need to sign up for them each separately (with Yahoo ID and Windows Live accounts, respectively).
Here are links to get you started with webmaster tools from the big 2 search engines: