On a demand curve when the demand increases the price will decrease.
You actually mean "along the demand curve, a decrease in price will increase quantity demanded, all else equal". This is the law of demand, and it holds for ordinary ("non-Giffen/Veblen") goods that have downward-sloping demand curves. There is no shift of the demand schedule in this scenario:
However on a demand and supply graph, when the demand shifts to the right, the price will increase.
Yes, when the demand curve shifts to the right (all else equal), the equilibrium market price rises to account for the fact that consumers are now willing to pay more. The increase in price is the mechanism by which excess demand at the initial price is cleared:
There is no contradiction here in the supply-and-demand model. Note that movements along the demand curve can only occur when we are away from a market equilibrium. These price movements are traced out by shifts of the supply curve and they continue until the new market equilibrium is reached.
Based on your vocabulary, you are likely misunderstanding the demand curve with actual quantity demanded.