Elastic demand is a type of demand that will rise or fall depending on the price of the good. For example, candy bars are an elastic demand. If the price of candy is around $1, most people will buy the candy and it will be high in demand. However, if that same candy bar's price rose up to $4, most people would not buy the candy.
Inelastic demand is the opposite. People will buy goods with an inelastic demand no matter what the price is. A good example of this would be gas. People complain and complain about gas prices, yet they still buy it because they need it, even if it $3 a gallon. Another example would be life-saving medications. Even if they are expensive, people will still buy them or else they could possibly die.
So basically the elastic demand for a good will rise or fall depening on the price where as an inelastic demand for a good won't.