What is ImageMagick? A no holds barred summary
ImageMagick is designed for batch processing of images. That is, allow you to create more complex methods and techniques in a script (shell, dos, perl, PHP, etc) so that you can then apply the image processing operations to many images, or as part of a sub-system of some other tool, such as a web application, video processing tool, panorama generator, and so on. It is not a GUI image editor.
ImageMagick is first of all an image-to-image converter. It was what it ws originally designed to do. That is it will convert an image in just about any image format (tell us if it can't) to any other image format.
But it is also a library of Image Processing Algorithms. These can be access via the command line, and shell/dos scripts, which is what these example pages are specifically about, or via a large number of programming languages, such as C, C++, Perl, Ruby, PHP, etc, etc, etc. (see ImageMagick API's)
Speed was however never a major feature, more the quality of its image results. That is not to say that it can't transform images in a reasonable time, but it is not blindingly fast. Because of this it can be slow for some processing operations, especially in its attempts to compress images into image formats that have limited capability.
ImageMagick however concerns itself mainly with images in the form of a 'raster' or "rectangular array of pixels". It will handle 'vector' image formats like Postscript and PDF, but at the cost of converting those images into a 'raster', and generating a vector image wrapper around a raster image. As a result vector images are often processed badly in the default case, but specific options can be used to improve this situation. See, A word about Vector Image formats.