I always begin lectures on elasticities by relating the idea to that of a rubber band. If the rubber band is inelastic (not very stretchy), then it does not react (stretch) very much when you exert effort to stretch it. This idea is very similar to what happens when looking at elasticities in economics. If you have an inelastic price elasticity of demand, then if you change price (similar to your effort), then the quantity demanded does not change very much (how much the rubber band stretches).
Lets now compare this example to the math we always see in the textbooks.
The above relationships should give you a general idea of how elasticities work, and how they are calculated, but why are they important?
Understanding the price elasticity of demand is very important for a business because it can tell them how much they can raise their prices before they start to lose too many sales. For example, say that the price elasticity of demand is inelastic, then they can raise the price a lot, and people will still buy similar amounts of the good! Think about medicine, or cigarettes. When the price of these things go up, people still want or need them so they are willing to pay the higher prices.
This also holds true for the government when decided what types of goods to tax. Taxing cigarettes and alcohol is very popular because the price increases do not change the quantity demanded to a large extent. Gasoline is also typically has an inelastic price elasticity of demand.
So what types of goods have elastic price elasticity of demands? Goods with a lot of substitutes, or things that we do not really need. Imagine if the price of a coffee at Starbucks went up to $10 for a venti. You would probably go to another coffee shop or brew your own at home. Likewise, if the price of blue pens went up, you would probably buy black pens instead. Because we react to the price increase by buying less of the good, we would consider our price elasticity of demand elastic (the rubber band reacts a lot to a little effort).