Continuously acquiring new knowledge from a dynamic environment is a fundamental capability for animals, facilitating their survival and ability to address various challenges. This capability is referred to as continual learning, which focuses on the ability to learn a sequence of tasks without the detriment of previous knowledge. A prevalent strategy to tackle continual learning involves selecting and storing numerous essential data samples from prior tasks within a fixed-size memory buffer. However, the majority of current memory-based techniques typically utilize a single memory buffer, which poses challenges in concurrently managing newly acquired and previously learned samples. Drawing inspiration from the Complementary Learning Systems (CLS) theory, which defines rapid and gradual learning mechanisms for processing information, we propose an innovative dual memory system called the Information-Theoretic Dual Memory System (ITDMS). This system comprises a fast memory buffer designed to retain temporary and novel samples, alongside a slow memory buffer dedicated to preserving critical and informative samples. The fast memory buffer is optimized employing an efficient reservoir sampling process. Furthermore, we introduce a novel information-theoretic memory optimization strategy that selectively identifies and retains diverse and informative data samples for the slow memory buffer. Additionally, we propose a novel balanced sample selection procedure that automatically identifies and eliminates redundant memorized samples, thus freeing up memory capacity for new data acquisitions, which can deal with a growing array of tasks. Our methodology is rigorously assessed through a series of continual learning experiments, with empirical results underscoring the effectiveness of the proposed system.