tool set

The Toolset
Google have provided a useful toolset to help diagnose site problems, and give you insights into how to perform better in Google's index.
Site Maps
Webmasters should submit a Google Sitemap to ensure their site is crawled.
Whilst not necessary, as Google should be able to crawl your site via links, it's a good idea to have the sitemap in case there are problems during the regular crawl. To create a Google sitemap, you can use Google's generator script. There are also various third party tools to help you do this.
Also see "Subscriber Stats" below.
Crawler Access
You can use this tool to test your robots.txt file. Check out this article about why you might use a Robots.txt. You can also use this tool to generate a robots.txt file, based on drop down options.
You can also use this tool to remove a URL from Google. Careful with that one ;)
Site Links
You can specify which site links you want to show. Sitelinks are the indented links below a listing in the search engine result pages.
Sitelinks look like this:

It's a good idea to specify pages that users will find most useful, and that display a consistent theme for your site. You can also block irrelevant pages from appearing in the site links using this tool.
Settings
If your site targets a specific country, you can specify this information. This is very useful if you have a .com, but target visitors in a country outside the US. The tool makes no difference if you already have a country specific TLD, however.
The preferred domain setting enables you to specify which URL you want crawled, the non-www version, or the www version.
Why would you do this?
www.acme.com and acme.com can be seen by crawlers as being two different domains. This can sometimes lead to duplicate content issues. So it's a good idea to pick one or the other, and specify it in Google tools.
The crawl rate setting allows you to slow down the search spiders crawl rate if you're having problems with the spider gobbling to many of your server resources. By default, Google sets the crawl rate for all sites.
Top Search Queries
Find out which keywords you rank for, which keywords send you the most traffic, and which results users clicked through on. If you rank for a lot of keyword terms, but aren't getting much click through, you might want to review the wording of your title tags.
You can also sort this data by device, such as mobile. You can find out which Google country domains send you the most traffic, as well as sorting by image search, and you can specify a date range.
Links To Your Site
You can find out which sites are linking to you, and what anchor text they're using. You can also download this data in the form of spreadsheets.
Keywords
Find out the most common keywords found on your site.
Are these words consistent with the theme of your site? If not, you need to balance out the content by adding more on-topic pages.
Internal Links
Shows which pages are most heavily linked internally, and from where. You can also see if pages are linked from non-www or www "domains". Aim to get internal linking consistent i.e. from either non-www or www but not both, and make sure your money pages are well linked.
Subscriber Stats
Find out how many people subscribe to your RSS feed via Google Reader. You can also submit a feed as a sitemap.
Crawl Errors
This useful diagnostic tool will show you which pages Google is having problems crawling. There is a breakdown by web, mobile, and mobile WML.
Crawl Stats
Use this tool to find out how many pages are crawled per day, and the time spent downloading those pages. This can reveal problems such as slow servers and page load errors.
Crawl stats also give you a PageRank distribution chart. However, like the toolbar, take this data with a grain of salt. I've found it bears little relationship to site ranking, and the real PR of pages is only known by Google.
HTML Suggestions
Use this tool to find duplicate and missing title and descriptions tags.