Two major deep learning methods for semantic segmentation, i.e., patch-based convolutional neural network (CNN) approaches and fully convolutional neural network (FCNN) models, are studied in the context of classification of regions in underwater images of coral reef ecosystems into biologically meaningful categories. For the patch-based CNN approaches, we use image data extracted from underwater video accompanied by individual point-wise ground truth annotations. We show that patch-based CNN methods can outperform a previously proposed approach that uses support vector machine (SVM)-based classifiers in conjunction with texture-based features. We compare the results of five different CNN architectures in our formulation of patch-based CNN methods. The Resnet152 CNN architecture is observed to perform the best on our annotated dataset of underwater coral reef images. We also examine and compare the results of four different FCNN models for semantic segmentation of coral reef images. We develop a tool for fast generation of segmentation maps to serve as ground truth segmentations for our FCNN models. The FCNN architecture Deeplab v2 is observed to yield the best results for semantic segmentation of underwater coral reef images.