A mobile application is a software developed to be installed on a mobile device, such as a smartphone, a tablet, a PDA or any digital device.
A mobile application can be either downloaded from an Application Store or directly loaded on the mobile device on the production line. The number of downloads are increasing sharply (more than 100 billion applications downloaded for AppStore & Google Play)
There are different mobile Operating Systems and the most popular are:
iOS from Apple running on iPhone and iPad,
Android from Google, running on many smartphones (Samsung, HTC, LG, Motorola...),
BlackBerry OS from Blackberry (ex-RIM),
Windows Phone from Microsoft,
Firefox OS from Mozilla.

Each of those OS has its own technology and its “store” from where the applications can be downloaded, either free or at a set cost.
Apple AppStore for iPhone and iPad,
Google Play for Android,
BlackBerry World,
Windows Store,
Firefox Marketplace.

A mobile application has many advantages compared to mobile web sites.
Here are 4 of them:
Ease and comfort of use and an unequaled user experience,
Direct access to the service by just clicking on the icon of the application standing on the dashboard of the smartphone or the tablet
Off-line usage
Integration of natives features, such as Push Notification for instance
It is possible to design and implement a “hybrid” application, based on native screens and web views HTML based screens (see "Native application vs. Hybrid application")