Oceanic eddies are a widespread and important occurrence that have a vital role in the movement of chemicals and energy within the marine ecosystem. Hence, the astute and precise recognition of these swirling currents may greatly contribute to the progress of our comprehension of oceanography. Due to the continuous breakthroughs in state-of-the-art deep learning technology, the population is witnessing a progressive improvement in the methods used to identify and understand these aquatic characteristics. This study employs Sea Surface Temperature (SST) data acquired from the Copernicus Marine and Environment Monitoring Service (CMEMS) in the Atlantic Ocean. The objective is to present EddyNet, a cutting-edge deeplearning framework specifically developed for the automatic identification and categorization of ocean eddies. EddyNet incorporates a pixel-wise classification layer into its neural encoder-decoder architecture. The resulting output is a map that maintains the same dimensions as the input, but each individual pixel is assigned a label indicating its classification as either "0" for non-eddy regions, "1" for anticyclonic eddies, or "2" for cyclonic eddies. We propose a new image segmentation method based on the U-Net architecture with different convolutional neural network backbones such as VGG16, VGG19, DenseNet121, and MobileNetV2. Our models are built and trained using Python and the Keras library with the Adam optimizer for improved convergence. Our approach uses sparse categorical cross-entropy as the loss function, simplifying the label encoding process for multi-class classification with sparse labels. Initial results show that this method achieves a good balance between computational efficiency and segmentation accuracy, making it suitable for realtime applications.