UPDATE: Due to changes made by Niantic Labs, tracking in Pokemon Go has changed dramatically. See our post on the current Pokemon Go tracking updates for more information. The Pokemon Go team at Niantic has done a great job at making tracking simple. Using the box in the bottom right corner, you'll see a listing of nearby Pokemon with each one having a set number of footprints attached to it. The footprints range on a distance scale of 0-3.
How To Track Pokemon using the Tracking Box
These footprints indicate how far the Pokemon are from you. For example, if you have a Pokemon with no footprint under it's very close to you. If it has three footprints under it, it's a good distance away from you. Therefore, you'll need to walk even further to try to find it. Think of the tracking system in Pokemon Go as a hot and cold system. 
There are two ways to go about using the tracking box: tracking a group or singling out one Pokemon. To track a group, click the tracking box and begin walking in a direction. When you see the number of footprints for a Pokemon drop, you're headed in the right direction. If it goes up, turn around and head around in another direction. To track one Pokemon, click the Pokemon in the tracking box that you want. Clicking the Pokemon icon will keep their image posted in the bottom right corner and make it easier to follow them. From there, follow the same hot and cold method as above. Below is a great chart of how to track a Pokemon. 
via hotlikesauce69
How to Use The Radar Ring to Track Pokemon
Pokemon Go features a radar ring that really makes things easy when it comes to tracking Pokemon and hitting up Pokestops. As you've probably noticed by now, there is a blinking radar ring round your Pokemon Go character. Think of this ring as your character's reach. For example, if your ring touches what you desire, you can probably attain it. This way you don't have to enter buildings all the time just to catch a Pokemon or reach a Pokestop. 
Using the insight the Pokemon tracker and radar ring provide, you can become the real Ash Ketchum in no time. If you see a Pokemon with no footprints and some rustling leaves around you, get your character's radar ring close to it and you should be golden. It's important to note that the blinking radar ring does NOT indicate a Pokemon is close to you. The same can be said for a blinking tracker box. When you tracker box blinks or flashes, it only means your query is updating.