You can plant seeds from your apple, but the resulting fruit from that seedling will not be exactly like the apple parent. Most apple seeds are formed when pollen from one apple cultivar pollinates the flower of another cultivar. This results in the combination of two different genetic parents and results in an offspring (seed) that has a different genetic code. The resulting seed and potential tree will have characteristics of both the pollen parent and the flowering parent. Therefore, it will produce different fruit. The common analogy is that children do not look exactly like one of their parents but rather have characteristics of both. This quality would hold true for any fruit crop that is cross pollinated by a different cultivar. With trees that are self-fruitful, seeds are formed by pollen landing on the receptive area of the flower of the same cultivar. These will have a greater tendency to produce trees that have very similar fruiting characteristics as the parent cultivar, but not 100% of the time. Peaches and nectarines are self-fruitful and therefore sometimes produce seeds that are very similar to the parent tree.
Answer provided by Dr. Robert Crassweller, Penn State University.