What is LEGO terminology for measurement of normal bricks and common scales? Where can I find basic LEGO terminology illustrated with LEGO parts, e.g. what's a stud? What's a plate? I also need to know the names of the most common scales of basic LEGO kits, and measurement of a most fundamental part(s) at each scale.
Terminology

This is the most basic terminology in a single image: 


A bit more verbose version is found in this slideshow, but there are a few factual errors, highlighted in the comments below.

The various parts are named based on their main characteristics, then their sizes in studs, as seen here: 

After getting the hang of it, it's time to advance to the more complex terminology, with the help of the indispensable blog New Elementary.

Measurements

The fundamental sizes for the basic bricks are seen in this answer, or a bit more simply: 

The height of the plates is 1/3 of the bricks, or 3.2 mm.

Scale

As for scales, this question is a bit complicated, as the basic reference unit is the LEGO Minifigure, which has weird proportions, resulting in inconsistent scales:

However, there are a few official ones, including:


minifigure scale (about 1:40)
microscale: anything way smaller than that, see these examples or these official ones.
midi-scale was only ever used officially for two sets, which are sized very differently, as both models are around 30 studs long, so it's 1:100 - 1:6000 which would definitely fall into microscale category.
towards the other end of the scale, there is life-sized, which of course does not get used much for official sets for sale, but rather for one-off projects: 
and it is definitely possible to create larger than life LEGO builds:  Or models of microscopical structures, like cells:  or even of subatomic scale: