Stress detection and monitoring is an active area of research with important implications for an individual's personal, professional, and social health. Current approaches for stress classification use traditional machine learning algorithms trained on features computed from multiple sensor modalities. These methods are data and computation-intensive, rely on hand-crafted features, and lack reproducibility. These limitations impede the practical use of stress detection and classification systems in the real world. To overcome these shortcomings, we propose Stressalyzer, a novel stress classification and personalization framework from single-modality sensor data without feature computation and selection. Stressalyzer uses only Electrodermal activity (EDA) sensor data while providing competitive results compared to the state-of-the-art techniques that use multiple sensor modalities and are computationally expensive due to the calculation of large number of features. Using the dataset collected in a laboratory setting from 15 subjects, our single-channel neural network-based model achieves a classification accuracy of 92.9% and an f1 score of 0.89 for binary stress classification. Our leave-one-subject-out analysis establishes the subjective nature of stress and shows that personalizing stress models using Stressalyzer significantly improves the model performance. Without model personalization, we found a performance decline in 40% of the subjects, suggesting the need for model personalization.