- Location of your readers
- The length of time that people are spending at your site
- Path of entry, exit, and everywhere in between
- How many visitors are new, returning; also how often they're returning
- What search engines people are using to find you
- The key words that people are searching to find you
- Number of total hits at any given time, as well as trends associated with this
- Number of unique visitors
Knowing this information can help you tailor your writings to appeal to the most people (for instance, I continue to post monthly "special days" since this is one of the top search ways I'm found), find out where you're losing people (I seem to have a following of people who will check to see if I'm having any giveaways, and if not, they leave!), determine what topics you're writing about that interest people the most (just see what pages they spend the most time looking at!), and it can help you find out what other sites are giving traffic to you.
The second way in which knowing your website's statistics is helpful is when you're looking to do any product reviews or offer advertising space. Many companies determine whether they want to work with someone or not partially based on that blog's traffic statistics. I'm commonly asked where my readers are located, and how many hits I receive per month.
The two best stat counters I've found for blogging and websites in general are Google Analytics at http://www.google.com/analytics/ and Stat Counter at http://www.statcounter.com/ . Both of these services are free and provide an in depth look at how your site is doing. For the beginner or general blogger, I'd recommend using Stat Counter. It provides an easy to read summary of your basic stats--numbers related to page loads, location of visitors, and traffic sources among other things. The graphical illustrations are easy to understand and can provide you with information that's quick to read. If you're hoping to find a deeper analysis of your site, then Google Analytics is for you. This site will provide you with details of all of the statistical analysis as well as comparing your goals and current statistics to all of your past ones.
I am registered at both Google Analytics and Stat Counter, but generally turn to Stat Counter since I normally just need a quick overview of things. If you have a blog or website and aren't already registered at one of these places, I think you'll find it to be worth your time and effort. The information you'll glean can be priceless in building a better blog!