Analysis of the placenta is extremely useful for evaluating health risks of the mother and baby after delivery. In this paper, we tackle the problem of automatic morphological characterization of placentas, including the tasks of placenta image segmentation, umbilical cord insertion point localization, and maternal/fetal side classification. We curated an existing dataset consisting of around 1,000 placenta images taken at Northwestern Memorial Hospital, together with their pixel-level segmentation map. We propose a novel pipeline, PlacentaNet, which consists of three encoder-decoder convolutional neural networks with a shared encoder, to address these morphological characterization tasks by employing a transfer learning training strategy. We evaluated its effectiveness using the curated dataset as well as the pathology reports in the medical record. The system produced accurate morphological characterization, which enabled subsequent feature analysis of placentas. In particular, we show promising results for detection of retained placenta (i.e., incomplete placenta) and umbilical cord insertion type categorization, both of which may possess clinical impact.