Demand-Pull inflation is caused when people have too much money to spend on a scarce amount of goods.
For example, if the government were to increase the minimum wage to 15 dollars and hour then everyone would have more money. But more money does not mean that there are more goods or resources to produce those goods with. In order to keep things balanced the prices of goods will increase until the buying power of the new minimum wage($15/hr) is about equal to the buying power of the old minimum wage($7.25/hr). Increasing the minimum wage would have many other effects as well but for the sake of the example its better to focus solely on inflation.
Cost-Push inflation is caused when rising costs drive up the prices of scarce goods.
For Example, if the cost of electricity increased, then the cost of producing everything increases as well. Increased production costs results in companies producing an inadequate amount of goods at the current price level. So, in order to restore production to normal levels, the price of the goods increases and production goes back up.
Looking at this another ways, lets assume that a company produces 100 widgets at at price level of 100. Suddenly the cost of producing widgets doubles and the profit margin for widgets takes a dive. Now the company is only will to produce 25 widgets at a price level of 100; however, everyone loves widgets and so the price level increases to 150 and the production of widgets increases to 80.