App is short for "application," which is the same thing as a software program. While an app may refer to a program for any hardware platform, it is most often used to describe programs for mobile devices, such as smartphones and tablets.
The term "app" was popularized by Apple when the company created the "App Store" in 2008, a year after the first iPhone was released. As the iPhone and App Store grew in popularity, the term "app" became the standard way to refer to mobile applications. Programs for Android and Windows Phone are now called "apps" as well.
Unlike applications for traditional PCs (often called "desktop applications"), mobile apps can only be obtained by downloading them from an online app store. Most devices automatically install apps when downloaded, which creates a seamless installation process for the user. Some apps are free, while others must be purchased. However, mobile apps are typically much cheaper than PC applications, and many are available for only 99 cents. In fact, most paid apps are less than $10.
Part of the reason mobile apps are cheaper than desktop applications is because they are often less advanced and take less resources to develop. Apps are limited to the capabilities of the mobile operating system (such as iOS or Android) and therefore may not offer as much functionality as a desktop program. For example, a word processor for Android will most likely have significantly less features than a word processing application for Windows. Most apps are designed to be small, fast, and easy-to-use. Unlike desktop applications, apps are intended to be used on-the-go and are developed to advantage of a small touchscreen interface.
NOTE: Apple released the Mac App Store in January, 2011, which offers downloadable applications for Mac OS X. In this case, the term "app" refers to desktop applications.