In the context of supply and demand discussions, demand refers to the quantity of a good that is desired by buyers. An important distinction to make is the difference between demand and the quantitiy demanded. The quantity demanded refers to the specific amount of that product that buyers are willing to buy at a given price. This relationship between price and the quantity of product demanded at that price is defined as the demand relationship.
Supply is defined as the total quantity of a product or service that the marketplace can offer. The quantity supplied is the amount of a product/service that suppliers are willing to supply at a given price. This relationship between price and the ammount of a good/service supplied is known as the supply relationship.
When thinking about demand and supply together, the supply relationship and demand relationship basically mirror eachother at equilibrium. At equilibrium, the quantity supplied and quantity demanded intersect and are equal.
In the diagram below, supply is illustrated by the upward sloping blue line and demand is illustrated by the downward sloping green line. At a price of P* and a quantity of Q*, the quantity demanded and the supply demanded intersect at the Equilibirum Price. At equilibrium price, suppliers are selling all the goods that they have produced and consumers are getting all the goods that they are 
demanding. This is the optimal economic condition, where both consumers and producers of goods and services are satisfied.
Very simply, the law of demand states that if all other factors remain constant, if a good's price is higher, fewer people will demand it. As the price of that good goes down, the quantity of that good that the market will demand will increase. In the diagram below, you see this relationship. At price P1, the quanity of that good demanded is Q1. If the price of this good were to be decreased to P2, the quantity of that good demanded would increase to Q2. The same is true for P3 and Q3. When prices move up or down (assuming all else is constant), the quantity demanded will move up or down the demand curve and define the new quantity demanded.
After understanding the law of demand, the law of supply is simple, it's effectively the inverse of the law of demand. The law of supply states that as the price rises for a given product/service, suppliers are willing to supply more. Selling more goods/services at a higher price means more revenue. In the diagram below, you can see that as the price shifts from P1 to P2, the quantity supplied of that good shifts from Q1 to Q2. The movement in price (up or down) causes movement along the supply curve and the quantity demanded will change accordingly.
