Ofcom sum up the answer to this question very well.

A mobile application (or app) is a computer programme which runs on a mobile handset.Mobile ‘apps’ have existed for as long as there have been smartphones, but the complexity of applications has been limited by the handset’s memory and processing power. In the last five years many handsets have had the ability to run simple mobile applications, but until recently ‘apps’ have not had a great impact on the mobile phone industry.

‘Apps’ exist across many genres, including games, entertainment, utilities, education, travel and lifestyle. Furthermore, they vary in the degree to which they rely on the capabilities of the handset they run on and the network they are connected to. For example, many games can run independently of data connectivity, while other utilities and travel applications can either take advantage of GPS built into handsets, or use location-based information provided by the mobile network operator.

Examples of applications from Apple’s All-Time Top Free Apps include: Facebook – an application which accesses the popular social networking site; Urbanspoon – a travel application which recommends restaurants in your local area; Shazam – a music application which identifies songs using the phone’s microphone; and Remote – an entertainment utility which can control the iTunes media player on a remote computer from an iPhone or iPod Touch. Ofcom Communications Marketing Report, 2009