are any software stored in a remote server that delivers its functions through the internet and a web browser. Usually, user enters the information through the web browser and it gets transmitted through the web. Then the server stores, retrieves and provides the information. The best example for this is email such as Microsoft's Outlook and Google's Gmail.
For the past few years, advances in technology have been pushing the development of web applications to have functions that won't require the need to store information on a remote server. Just like a Word Processor, an application that doesn't require a remote server and can store files on the computer. So in other words, there's a push to have web applications that can store info and data on cloud and allow users to download them anytime on their computers.
Those who have been using the web for a long time have witnessed how well known applications such as Yahoo Mail, Gmail and others have evolved. They have seen how web applications have become more sophisticated. It became more sophisticated because of AJAX. AJAX is a programming model used for creating responsive web apps.
Mobile applications like Facebook and Twitter app are also prime examples of the evolution of web applications. They have been designed to cater to needs of mobile users.