A computer cluster is a group of connected computers that work together closely like a single computer. Clusters allow for greater computing power than a single computer by distributing workloads across nodes. They provide improved speed, reliability, and cost-effectiveness compared to single computers or mainframes. Key aspects of clusters discussed include message passing between nodes, use for parallel processing, early cluster products, the role of operating systems and networks, and applications such as web serving, databases, e-commerce, and high-performance computing. Challenges also discussed include providing a single system image across nodes and efficient communication.