A client application (also known as a client app) is a piece of software (an application) that is installed on a user's local computer (also known as a client). In terms of SharePoint and its related software, "client application" usually refers to any of the Office suite of software that offers both an online, browser-based version and a corresponding installed version that exists on the user's computer.
Outlook, Word, Excel, PowerPoint, and OneNote are the most common applications that offer both client application and browser/online versions; Office Online (also known as Office Web Apps in many on-premises versions of SharePoint). Additionally, Visio and Project provide an online viewer that allows users to open and read those files. Adobe Acrobat is another example of such software; you can open and read PDF files in the plugin for most browsers or you can open PDFs directly in the Acrobat Reader, the free PDF reader provided by Adobe.
Generally speaking, the client application provides more functionality than the online version of an application. This is usually due to limits in the sophistication of writing an application for a browser versus stand-alone applications. Another limit is load time and bandwidth usage; if the online app takes too long to load or eats up too much internet bandwidth, the utility of the app can quickly be diminished. Online apps are generally good for quick, uncomplicated access or editing.
In the case of Microsoft Office, generally the online version of the app loads more quickly than the client app due to the online app being a simplified version of the client app. This of course depends on your internet connection, but it's a useful test to run both for your own sake and to share the test results with users who may be hesitant to adopt the online version of the application.