Today I was trying to remember how to use Young's Modulus. I eventually figured it out (I think), but when I went to check my intuition, nobody else was describing it in these terms. So it's odd that this wouldn't be a well-known intuitive aid, which I guess means I might just be wrong about it. If so, please point that out in the comments.
But the mental shortcut is this: Young's Modulus (aka the Elastic Modulus) tells us how much pressure we have to apply to a material to double its length.
For example: Aluminum has a modulus of elasticity of 10 million PSI. So if you had a piece of aluminum bar stock 1 inch square, and you pulled on it with a force of 10 million pounds, it'd end up twice as long as it started.
Now, in reality the bar would tear before it'd stretch that far, because the maximum elongation for aluminum tends to be less than 30%. But remembering that "E = pressure required to double the length" gets you into the right place to use the units correctly (for example, to say that 1% of 10 million PSI would elongate our bar by 1%).