Technorati is a well-known website that aims to search, index, and rank the estimated 60 million blogs around the world. When a blogger starts out, your ranking on Technorati tends to be on the low side, if you have one at all. Like Alexa, the ranking is reversed: the smaller the number, the better the ranking. As you move up (like I did), the ranked number decreases as the number of links and blogs linking to your blog increase.
You can build up your ranking in several ways: by linking to blogs and by having blogs linking to yours (called inbound links):

Technorati reads all of the HTML code in a blog posting, and also tracks all of the activity around a blog or post such as inbound and outbound links.

There are a variety of ways that such links can be established: the most obvious are comments, trackbacks, blogroll links, links in posts, perhaps even text links such as ads. All of these can act to increase your ranking. Additionally, I have heard that including a blog in your favorites at Technorati and others including your blog in theirs has a positive effect, too.

So, how do you do it? Basically,

1. comment on top bloggers postings (thanks, John Chow!),
2. link to other articles,
3. encourage people to link to your blog and your articles,
4. exchange favorites in Technorati,
5. use your Blogroll to exchange links.

There are more good strategies I found here, though they are perhaps less orthodox.

Naturally, as John Chow found out, there are ways to be banned at Technorati so avoid using blackhat techniques that may risk a hard-earned reputation! Of course, I had problems just claiming my blog, I kept getting an error every time that I tried to claim my blog. And, initially, writing a request for help took a while to solve.