To be precise, a good itself cannot be elastic in economic terms. Instead, it is proper to say that demand for the good is elastic. When we say that demand for a good is elastic, we mean that, when the price of the good goes down, the revenue gained by selling that good goes up.
The law of demand tells us that, when the price of a good goes down, the quantity of that good that is demanded goes up. But it does not tell us how much the quantity demanded goes up. This is where elasticity comes in.
If the price of a good is lowered, it is possible that many more people will buy it. It is possible that so many more people will buy it that the seller ends up making more money. For example, let us imagine that I sell hot dogs at a stand for $2 each. I sell 10 per hour and make $20 per hour. Now let us imagine that I drop the price to $1.50 and I sell 15 per hour. I now make $22.50 per hour. The demand for my hot dogs was price elastic because I lowered the price and sold so many more hot dogs that I ended up making a larger amount of revenue than I did when the price was high.
So, when we talk about the price elasticity of demand, we are talking about how much more of a good people will buy if the price drops (or how much less they will buy if the price rises). We say that the demand for a good is price elastic if the price drops and the quantity demanded rises enough that the seller actually gets more revenue.