Analyzing web traffic involves collecting, calculating, visualizing and critically thinking about the data generated by visitors' web browsers while they are viewing webpages. The resulting data provides insight into how visitors use the site and why they stay or leave. The key concept to analytics is learning about your users so you can improve your web application to better suit their needs.
It's easy to get overwhelmed at both the number of analytics services and the numerous types of data points collected. Focus on just a handful of metrics when you're just starting out. As your application scales and you understand more about your users then add additional analytics services to gain further insight into their behavior with advanced visualizations such as heatmaps and action funnels. The seven stages of startup analytics grief post is an amusing read and provides context for how to begin and then grow tracked metrics over time.
If your application is selling a product or service you can ultimately build a user funnel (often called "sales funnel" prior to a user becoming a customer) to better understand why people buy or don't buy what you're selling. With a funnel you can visualize drop-off points where visitors leave your application before taking some action, such as purchasing your service.
Google Analytics is a widely used free analytics tool for website traffic.
MixPanel's analytics platform focuses on mobile and sales funnel metrics. A developer builds what data points need to be collected into the server side or client side code. MixPanel captures that data and provides metrics and visualizations based on the data.
KISSmetrics' analytics provides context for who is visiting a website and what actions they are taking while on the site.
Heap is a recently founded analytics service with a free introductory tier to get started.
CrazyEgg is tool for understanding a user's focus while using a website based on heatmaps generated from mouse movements.
Pandas and Google Analytics shows how to use pandas for data analysis with Google Analytics' API to perform calculations not available in the tool itself.
This beginner's guide to math and stats behind web analytics provides some context for understanding and reasoning about web traffic.
This post provides context for determining if a given metric is "vanity" or actionable.
Read this post on how your analytics software actually works to get a better understanding of what's going on behind the scenes from a technical perspective.