What is Java and why do I need it?

In short, Java is a programming language and a computing platform. It is free to download and available for a large number of operating systems, like Windows, Linux or Mac.
So, what kind of problem does it solve? When someone writes a computer program, it has to be written for the kind of computer and operating system that it is targeted for. When writing a program intended to run on an Intel processor using Microsoft Windows as an operating system, this program will run happily on that particular system, but it won't work on a MAC for instance. That's not really helpful if someone wants to write a program for a wider audience. The same program had to be written and compiled in multiple versions to cater for all the differences on a large number of systems.
And that is the problem, Java tries to solve. The Java Runtime Environment (JRE) is a program that serves as a basis for Java programs. Think of it as a kind of layer between your computer and the program that you want to run:
Layering of software
Layering of software
In order to run Java programs, the user installs a JRE for their particular platform. If you are for instance working on 64 bit Windows, you'll probably download jre-8u45-windows-x64.exe and double-click that file to start the setup process. After installation any Java program can be executed, not just PEE Mail.