How do I properly reference? I’m writing an essay, and have been asked to use Harvard Referencing.
From what I know this type of referencing uses citations similar to (my brain, 2016).

And then contains a bibliography like:

Bibliography  — or — References

My Brain (18 Dec. 2016) - “Title of article, or something” available from my skull

...

I've seen referencing done with [number's] instead of (author / publisher, date) in a few papers and - probably most famously -  wikipedia.

Is this not classed as Harvard referencing?
I find it so much neater and easier to follow, especially if I'm going to refer to multiple peices of work by the same author/publisher.

Also, I've noticed some papers placing mini bibliographies at the end of every page, containing all the references for that page.

Is this not harvard referencing?

I'd like to do this as it removes the pain of having to go all the way to the bibliography to check a citation.
The Harvard citation system does not use reference numbers.  It uses, as you describe, the author's last name and publication year.  (This is similar to the APA citation style.)

The reference lists should be at the end of the document, not listed as footnotes.  So your understanding of the Harvard style seems to be basically correct.  Some more details about the Harvard citation style are given here: http://www.citethisforme.com/harvard-referencing 

Personally, I too prefer referencing by number rather than author, but if you are supposed to use the Harvard style, then you just have to use it.