The short answer is that a Web Application is a piece of software that uses web servers, the internet and your browser to complete tasks. A web application is different to a web page in that you can typically interact with it by sending it data.
How do they work?
Almost all modern web applications split their work between your web browser and a web server. Typically they deliver a page, some code, design files, images and video to your web browser which then makes sense of it all to present a screen of information and controls to you. Usually when you click on some control, your browser sends a request to the server for more information. This request may contain information that you've filled out on a form. When the server receives this request, it uses more code to process and organise this information
A well designed web application does as much of its work as possible in your browser to avoid the time it takes to have a conversation with the web server. So, for example, when you fill out a form containing your email address, the browser will check that this is a valid email address before sending it to the server.
Where do I get them?
The most convenient (and perhaps scary) thing about web applications is that you don't even know that you're using them. Any website that uses a shopping cart, a discussion board, a comments section or a form that you submit your email address has a web application.
If you've used Facebook, Twitter or Gmail in a web browser, you've been interacting with a Web App without even knowing it. These companies all provide powerful web applications that allow their customers to perform complex tasks without even really needing to install any software.
How are they different from Mobile Apps?The key difference between mobile and web apps is that you need to install a mobile app from an App Store while web apps just run as part of a web page. So it turns out to be a lot easier to get started on a web app. However, some advanced features are difficult if not impossible to implement on a web application. So in the long-term if you keep returning to a web application, it may be better to use mobile apps instead.
Mobile Apps make complex features like notifications, location tracking and video a lot simpler to use.
Does my business need a Web Application?
If your business's web presence is purely about a providing a catalogue for your customers, you don't need a web application. However, if you'd like to use the web to start a conversation with your customers, you're going to need some form of web application to manage that.
There is a school of thought out there that your business needs a mobile app to serve their customers. But if you want people to interact with you, it is usually best to build a web application first so that people can use it by going to your website. If it turns out that you've got a lot of people returning all the time, it may be better to build a mobile application later for your most loyal customers.