Can you please explain to me the difference between Desktop Down PX Based, Desktop Down EM Based, Mobile First PX Based and Mobile First EM Based?

Desktop Down PX Based = Designing left to right using Pixels.
Desktop Down EM Based = Designing left to right using EM's.
Mobile First PX Based = Designing right to left using Pixels.
Mobile First EM Based = Designing right to left using EM's.

What are EM's?

http://kyleschaeffer.com/development/cs … -vs-pt-vs/

It is a scalable size unit for text. One 'em' is the size of the letter m. Ems are used for responsive design instead of px.

I've searched for hours, probably days, for the answer to this question, and merely renaming the style of layout is entirely unhelpful. The very first decision we need to make before every other thing is decide,
'Mobile up' or 'Monitor down', I can conceptualise that. EM or PX though?, why would I choose which? What are the advantages and disadvantages with RLM. Which is easier, which is better, and why and how? file-->new from--> Can anyone help me get started?

This article will explain things.
http://alistapart.com/article/howtosizetextincss

Another interesting article is here

Read this article first
http://www.narga.net/understanding-font … rcent-rem/

Then watch this video
https://www.youtube.com/watch?v=r7LGKtnEWmk

Jacques, some people prefer to start with a phone layout first (basically one column), and then rearrange that content as the screen width allows. Others like to start with the full-size site and work it down to one column.
Starting with the phone version first will get you thinking about what what's really important as far as how the content is prioritized and the site is navigated.
Here's a good responsive design overview: http://blog.teamtreehouse.com/modern-fi … web-design

As for px vs. ems (or rems), I prefer ems/rems for text, ems/rems and or percentages for layout and just use pixels for borders, horizontal rules and sometimes vertical spacing.
Conversion chart: http://pxtoem.com/
(Left panel displays the base pixel measurement, right panel displays the em sized based on it. 16px=1em or 10px=1em are the most common choices).