Accurate uncertainty information associated with essential climate variables (ECVs) is crucial for reliable climate modeling and understanding the spatiotemporal evolution of the Earth system. In recent years, geoscience and climate scientists have benefited from rapid progress in deep learning to advance the estimation of ECV products with improved accuracy. However, the quantification of uncertainties associated with the output of such deep learning models has yet to be thoroughly adopted. This survey explores the types of uncertainties associated with ECVs estimated from deep learning and the techniques to quantify them. The focus is on highlighting the importance of quantifying uncertainties inherent in ECV estimates, considering the dynamic and multifaceted nature of climate data. The survey starts by clarifying the definition of aleatoric and epistemic uncertainties and their roles in a typical satellite observation processing workflow, followed by bridging the gap between conventional statistical and deep learning views on uncertainties. Then, we comprehensively review the existing techniques for quantifying uncertainties associated with deep learning algorithms, focusing on their application in ECV studies. The specific need for modification to fit the requirements from both the Earth observation side and the deep learning side in such interdisciplinary tasks is discussed. Finally, we demonstrate our findings with two ECV examples, snow cover and terrestrial water storage, and provide our perspectives for future research.