Xposed Installer

This is the installer for the Xposed framework, which is a requirement for all modules.
Xposed is a framework for modules that can change the behavior of the system and apps without touching any APKs. That's great because it means that modules can work for different versions and even ROMs without any changes (as long as the original code was not changed too much). It's also easy to undo. As all changes are done in the memory, you just need to deactivate the module and reboot to get your original system back. There are many other advantages, but here is just one more: Multiple modules can do changes to the same part of the system or app. With modified APKs, you to decide for one. No way to combine them, unless the author builds multiple APKs with different combinations.

Note that this only works with root access on Android 4.0.3 or later.

Step 1: Enable Unknown Sources

First up, since Xposed is not available on Google Play, you'll have to make sure your phone allows for installation of sideloaded apps.
To do that, simply head to Settings and tap Security. In here, tick the box next to Unknown Sources and press OK.

Step 2: Download the Xposed Installer App

The Xposed Installer app is what allows you to install the framework upon which modules can call to make their system-level changes. It also provides a nice interface for searching and downloading the modules.
To get the latest version of Xposed Installer, just tap this link from your phone's web browser.
Once the download has finished, tap the notification to launch the install process. From here, hit Install, then when finished, tap Open.

Step 3: Install the Xposed Framework

Now that you've got the Xposed Installer app taken care of, the next step is to install the actual framework that Xposed modules use to work their magic.
From the Xposed Installer's main screen, just tap Framework to get started. You'll immediately see a warning that lets you know you should Be careful. Tick the box next to Don't show this again and press OK.

Next, tap the Install/Update button, and Xposed will ask for Superuser privileges. Grant it those, then you'll see a message that tells you that the installation was successful and a reboot is needed. Tap OK here, and your phone will automatically reboot.

Step 4: Installing Modules

Now that you're back up, you are ready to begin installing the modules that offer Xposed's main functionality. Just open up the Xposed Installer app and head to the Download section to have a look at what's available.