This is going to be a relatively quick post—not just because it's Friday, but because if you have rooted your Android phone, or even know what rooting means, the app I'm going to pitch here doesn't need too much explanation, beyond that a rooted Android phone is required.
Greenify is an app with a name that makes it sound like it will prevent your phone from spewing some kind of tree-killing gas. Close, but not quite. What Greenify is, really, is a list of your apps: apps that are running in some form right now, apps that might run if some condition (charging, Wi-Fi, etc.) changes, and apps that just check in every so often. All these things cause your phone to "wake up" from sleep, and, especially if they use the internet to check on things, cost you battery life.
So you install Greenify, launch the app, and look at what's running, and what will run at some point.
Think about the apps that you need running in the background. Your messaging app, Google Voice, Hangouts? Sure. Facebook and Twitter and Google+, if you really like getting notifications of new activity? Indeed. Your online banking app? Not quite. An image editing tool? You would be more than a little surprised at the variety and number of apps that try to do stuff when you're not looking at them. Almost none of them are malicious, but some of them are going to cost you battery life and response performance.
So tap on the apps that Greenify shows you as running, but you can't quite figure out why, and the apps that it knows will try to wake up whenever certain things happen. Hit the checkmark button in the upper-right corner, and those apps are "hibernated," or, basically, told with a stern voice not to wake up unless called upon. Give your phone a few days, see how it feels, and continue to check in with Greenify every so often. If you make a mistake, open the list of "More Apps," choose the apps to restore, and un-hibernate them.
There's a lot more to be said about Android's system, and how it does and does not fairly allow apps to wake up and do things, and whether it's ridiculous to use a root-required app to trim some of them down. But for now, let's just agree that Greenify is an interesting approach to battery life troubleshooting, and very simple to use. Once you're past that whole endangering-your-phone-to-root thing, anyways.