Analytics can be utilized to show areas that are doing well and areas that are doing poorly on your site. It is important to remember that the software is showing you numbers, not causes. The cause of valleys or peaks in your metrics is up to you (and your team) to determine. However, more data points is helpful in narrowing down these causes.
There are many important metrics to be aware of when you are looking at your statistics and trying to come up way to motivate participation. I've got a few examples below pulled for various communities I am involved in (and I am willing/able to share).
Visits by local time vs Visits by Server Time
The time of day that your community members visit is important. Are they visiting during lunch when they likely have limited time to participate or are they visiting during a time when they are likely to have more free time and can engage in lengthy discussions? Related to this, are all of your visitors showing up at the same time or are they spread throughout the day? This is important, because if you have visitors that only show up when the site hasn't had activity for a while, they may get the impression that the community has few active users.
This graph is showing visitors by local time. To me, this is showing that community members are spending lunch time and evenings on the site. Few people are visiting during the night.
This graph, however, is showing visitors by server time. I am seeing that my (known) global community is visiting through out the day. This around the clock interaction cycle keeps the content active so that no one is visiting and seeing long periods of inactivity.
Length of Visits
Another important metric, to me, is length of visits. When users show up on the site, how long are they active on the site? Do the users pop in to see if anything is new and then leave, or are they spending long periods of time on the site?
This is showing that many visitors to this community pop in for less than 2 minutes. Likely they are checking if there are forum threads they are interested in. If they don't find one, they move on. However, the blip around the 15-30 minute bin shows that when users do find something they stick around for a bit.
Visitor Browser / Devices
Perhaps you are thinking of site changes and want to see what your users are using to visit. There are ways to see what browsers your users are using. This is important if you are planning on making changes that certain browsers don't support. If a majority of your visitors use that browser, your new feature will be worthless.
Here we can see that nearly 75% of the community uses Chrome (makes sense, as I took this from a technical community). There are also multiple mobile browsers listed in the browser list.
With browsers, we can see what features users have. Again this is important if we want to make changes. We can't utilize Flash if only a small minority have it enabled.
Here we can see that almost all of the visitors to this particular community have Flash enabled. 2/3s have Silverlight enabled and no one is using Real Player.
Incoming Search Terms
Many analytic packages offer the ability to see what search terms lead a user to your site. This type of information is important if you are planning on optimizing your search engine ranking. It also helps to see how you are being discovered.
Statistics about the site
Simple page analytics may not be enough to help you though. Does it show the whole picture? If you have a database, you have more information (you just don't know it). One example, using Moderators.SE as the site, is to see how healthy the site it. Stack Exchange provides access to a portion of their data using the Data Explorer. Using this, we can get very interesting information.
This shows me that the public beta for Moderators opened very strong, but has been gradually leveling off over the past few weeks. Then, there was a blip. Why was there this blip? The data doesn't tell me why. This comes back to one of my first points - it is up to you and your team to figure out why the data is the way it is. In this case, I suspect it is because we had two Hot Questions across the Stack Exchange network that week.
The stats from above are pulled from various sites.
These are by no means a comprehensive list of metrics, but they do provide a snapshot of how your community is growing.