Emacs is an IDE.
edit: OK, I'll elaborate. What is an IDE?
As a starting point, let's expand the acronym: Integrated Development Environment. To analyze this, I start from the end.
An environment is, generally speaking, the part of the world that surrounds the point of view. In this case, it is what we see on our monitor (perhaps hear from our speakers) and manipulate through our keyboard (and perhaps a mouse).
Development is what we want to do in this environment, its purpose, if you want. We use the environment to develop software. This defines what subparts we need: an editor, an interface to the REPL, resp. the compiler, an interface to the debugger, and access to online documentation (this list may not be exhaustive).
Integrated means that all parts of the environment are somehow under a uniform surface. In an IDE, we can access and use the different subparts with a minimum of switching; we don't have to leave our defined environment. This integration lets the different subparts interact better. For example, the editor can know about what language we write in, and give us symbol autocompletion, jump-to-definition, auto-indentation, syntax highlighting, etc.. It can get information from the compiler, automatically jump to errors, and highlight them. In most, if not all IDEs, the editor is naturally at the heart of the development process.
Emacs does all this, it does it with a wide range of languages and tasks, and it does it with excellence, since it is seamlessly expandable by the user wherever he misses anything.
Counterexample: you could develop using something like Notepad, access documentation through Firefox and XPdf, and steer the compiler and debugger from a shell. This would be a Development Environment, but it would not be integrated.