Twitter Analytics is now available to any user, not just Twitter's advertising partners. Christopher Penn discovered this, and wrote about it on his blog, and it has since been picked up by several tech news outlets.
You should be able to access Twitter Analytics by going to settings, Twitter Ads, and from the Ads homepage, going to "Analytics". From there, you can find "Timeline activity".
From there, Twitter will show you your history of tweeting, along with engagement metrics for each tweet: number of favorites, number of retweets and number of replies. You can organize the tweet chart by Best, Good, and All.
It will also show you a visual timeline of the mentions, follows and unfollows you receive over a two-month period.
If you select "Followers" from the dashboard, you can see a separate dashboard showing your follower activity over time. It shows most unique interests and top interests of your followers by percentage, as well as follower locations, including the top cities, and gender. Additionally, it shows who your followers also follow.
While the offering appears to be available to all Twitter users, it should be noted that the data isn't completely there for everybody.