Android is a Linux-based operating system designed primarily for touchscreen mobile devices such as smartphones and tablet computers. Initially developed by Android, Inc., which Google backed financially and later bought in 2005, Android was unveiled in 2007 along with the founding of the Open Handset Alliance: a consortium of hardware, software, and telecommunication companies devoted to advancing open standards for mobile devices. The first Android-powered phone was sold in October 2008.
Android is open source and Google releases the code under the Apache License. This open source code and permissive licensing allows the software to be freely modified and distributed by device manufacturers, wireless carriers and enthusiast developers. Additionally, Android has a large community of developers writing applications ("apps") that extend the functionality of devices, written primarily in a customized version of the Java programming language. In October 2012, there were approximately 700,000 apps available for Android, and the estimated number of applications downloaded from Google Play, Android's primary app store, was 25 billion.
CyanogenMod (pronounced sigh-AN-oh-jen-mod), is a customized, aftermarket firmware distribution for several Android devices (See above for supported devices & how to install CyanogenMod on said devices). Based on the Android Open Source Project, CyanogenMod is designed to increase performance and reliability over Android-based ROMs released by vendors and carriers such as Google, T-Mobile, HTC, etc. CyanogenMod also offers a variety of features & enhancements that are not currently found in these versions of Android.
MIUI developed by Xiaomi Tech, is an aftermarket firmware for cell phones based on the open-source Android operating system. It features a heavily-modified user interface that does away with the Android app drawer and has drawn comparisons with Apple's iOS and Samsung's TouchWiz UI. The Custom ROM includes additional functionality not found in stock Android, including toggles on the notification pull-down, new music, gallery, and camera apps, and an altered phone dialer that displays matching contacts as a user enters a number.