The relationship between these things is that supply and demand work together to determine the price of a good or service. This will happen unless there is some sort of government intervention that prevents the price from moving according to supply and demand.
In general, there is an inverse relationship between the supply of a good or service and its price. That is, when the supply of a good goes up, its price goes down. This relationship assumes that all other factors (including demand) do not change.
Demand works the other way -- it has a direct relationship with price. When demand increases (all other things being equal) price increases.
If both supply and demand move at the same time, then the impact on price will depend on which way the two move and how much they move in comparison to one another. For example, large increase in supply coupled with a small increase in demand will still lead to a drop in price.
As the demand and supply of a good or service increase and decrease, the price of that product changes as well. In this way, supply and demand determine what the price of that product will be.