To date, batteries are the most widely used energy storage devices, fulfilling the requirements of different industrial and consumer applications. However, the efficient use of renewable energy sources and the emergence of wearable electronics has created the need for new requirements such as high-speed energy delivery, faster charge–discharge speeds, longer lifetimes, and reusability. This leads to the need for supercapacitors, which can be a good complement to batteries. However, one of their drawbacks is their lower energy storage capability, which has triggered worldwide research efforts to increase their energy density. With the introduction of novel nanostructured materials, hierarchical pore structures, hybrid devices combining these materials, and unconventional electrolytes, significant developments have been reported in the literature. This paper reviews the short history of the evolution of supercapacitors and the fundamental aspects of supercapacitors, positioning them among other energy-storage systems. The main electrochemical measurement methods used to characterize their energy storage features are discussed with a focus on their specific characteristics and limitations. High importance is given to the integral components of the supercapacitor cell, particularly to the electrode materials and the different types of electrolytes that determine the performance of the supercapacitor device (e.g., storage capability, power output, cycling stability). Current directions in the development of electrode materials, including carbonaceous forms, transition metal-based compounds, conducting polymers, and novel materials are discussed. The synergy between the electrode material and the current collector is a key factor, as well as the fine-tuning of the electrode material and electrolyte.