It depends on the flower! Some flowers are incompatible with themselves, so no fertilization will happen. Self pollination is not very common, but it does happen.
There are two types. Type I is when the pollen fertilizes a carpel in the same flower. Type II is when pollen fertilizes a different flower on the same plant. Self pollination helps for rapid growth of a population, but can decrease fitness, because all individuals in a population are genetically similar. If a pathogen moves through the population, it could kill all of the individuals, because none vary to have resistance.