Stagflation is the occurrence of both stagnation, which is slowing growth and production levels, and inflation, which is the increase of the average cost of goods.
If production costs rise for some reason, such as higher oil prices, it can cause economic growth to slow down and the supply of goods in the market to drop. This is known as stagnation. The weakened supply of goods in the market and the higher production costs of the goods will cause the retail prices of the good in the market to go up.
When prices go up on most consumer goods, this is called inflation. Some inflation is good, because it can mean that more people are spending more money and are willing to pay slightly higher prices for goods.
When inflation is matched by increases in wages and so forth, the economy is seen to be doing well. However, if production and transportation costs rise, perhaps due to a spike in oil prices, companies might have to cut back on wages or fire workers.
Now there is less demand for the same goods whose prices are inflating anyway, and become more unaffordable to the workforce who is having a harder time making money. Stagflation is this situation in which stagnation and inflation are both increasing.
It is hard for a government or central bank to fix, because monetary policy interventions which slow inflation will make the stagnation situation worse, and interventions meant to get the economy out of stagnation tend to increase inflation rates.