"Apps" (short for applications) is the equivalent of "programs", the original term used to describe a set of instructions (code) that runs on a computer to perform a specific task.
Originally, the only way on a computer to run a program (or application) was to install that program on your computer. Later, the Internet came along and you could perform tasks via an Internet based application on a server somewhere and it is accessed through a web browser.
When smartphones and tablets came along, this was the environment that existed. These devices are simply computers like those that took up a large room but in a very more compact form.
As to the OP's question, installing an app on a smart phone or tablet means you are downloading and installing the code for that task on your device rather than perform that task using an application/program on a remote server somewhere via a web browser. This allows that application to be tailored to the functions available on your device rather than operate within the constraints of a web browser application that must operate on all the platforms that a web browser can run on. In many ways that is a plus but in the case of privacy and security it can be a minus because the app can access functions of the device not available to a web browser although this distinction is somewhat muddied in the case of cloud computing.