It's easy to track simple numbers of user sessions, but who are these people? Where do they come from, do they come often, and what else can we find out about them?
Google Analytics
Google Analytics is a free service from Google that tracks all sorts of info on your website. Many stats packages offer similar services but Google offers it for free. All it requires is embedding a bit of Javascript in every page of your website (for blogs this probably means once in the footer of your template).
We like it because it offers stats we don't normally get with our normal stat package, namely geo location of visitors and new vs returning visitors.
Visitor Geo Location
One nice feature of Google Analytics is their Geo Location map. Below you see an example of a geo map representing the top 500 locations in the world accessing walkerart.org:

Top 500 locations in the world accessing walkerart.org
This can help give a nice overview of where people are accessing your site from. There is also a detailed listing with percentages you can drill down to. This can give you a better idea of who is reading your blog. Perhaps your blog audience is stuck in the US and you want to branch out. That may mean posting more content with international appeal. Or, perhaps that's exactly what you wanted, which means you should continue doing what you're doing.
New vs Returning Visitors
Another nice metric is the number of new vs returning visitors you have. Building an audience and a community around your blog means having people want to come back to it. But it also means bringing in new people.

New (blue) vs Returning (yellow) Visitors to walkerart.org
Google Analytics also tracks this and perhaps does it better than normal log parsing software. That's because Google is able to set a cookie in your browser to know you are you. If you change I Ps? it will still know who you are, and if you're in an office that uses the same external IP for everyone it is able to single you out individually among all the computers in use. It's just more accurate.
Standardizing Stats
One thing many institutions do is of course compare their success to other institutions. The problem however is that we're all using different stats packages which run on different OS's with different preferences that are all tweaked differently. The comparisons are good, but not always accurate because of these differences.
If everyone used a solution like Google Analytics, it would help us all standardize to a set stats package in which all institutions could compare their numbers accurately, because we'd all be using the same analytics.
This does not have to replace an old stats package or normal stats reporting. We use both methods at the Walker. We don't report the GA numbers usually, but would certainly find it useful for comparing with other institutions.
The main downside of going the GA route is embedding it on every page. This is very easy on dynamic site that has a consistent section on the entire site (say a navigation), but one off sections, or static parts of a website it becomes less manageable. All blogging software is template based however, and does not pose this problem. Adding GA to a blog is relatively trivial.
Surveys
The best way to get info on your visitors is of course to ask them about themselves, as explained in the next section.