What causes the yellow leaf spots that later develop a brown center on my apple leaves?

This a fungal disease called cedar-apple rust. Cedar-apple rust is a common disease of apple and crabapple. The fungus that causes the disease, Gymnosporangium juniperi-virginianae, is unusual because it must spend a phase of its life cycle as a parasite on Juniperus species, such as red cedar or ornamental junipers. Control: Grow resistant apple or crabapple varieties. Apple varieties that normally show good to excellent resistance to cedar-apple rust include ‘Red Delicious’, ‘McIntosh’, ‘Arkansas Black’, ‘Winesap’, ‘Mollies Delicious’, ‘Spartan’, ‘Priscilla’, ‘Liberty’, and ‘Empire’. Varieties that are usually highly susceptible include ‘Prima’, ‘Sir Prize’, ‘Lodi’, ‘Jonathan’, ‘Rome’, ‘Golden Delicious’, and ‘Jonafree’. Destroy nearby wild, abandoned, or worthless apples, crabapples, cedars, and junipers. Although apples may receive a few spores produced on cedars several miles away, most spores come from cedars within a few hundred feet from the apple trees. Follow a recommended fungicide spray schedule, beginning at the pink-bud stage of growth and continuing every 10 to 14 days through the first or second cover spray. The cedar-apple rust fungi have usually exhausted their spores by this time. Contact your local Cooperative Extension office for fungicide recommendations.