What makes a lens a “fast” lens?

This term is often used and can be confusing to new photographers. It’s important to bear in mind that this is an old term and has been in use for many years, specifically, before the digital photography and electronic camera age.
In photography terms like this are often used inappropriatedly, by mistake, or, simply overlapped with similar terms.
In the context of this subject I will give a quick overview of what it actually means.
Remember: When referring to fast lenses, photographers (including myself) use terms like “wide aperture” or “larger aperture”, meaning the same thing. So, in this context “wide or wider”, refers to the aperture, not the focal length of the lens.