A long-standing proposition is that by emulating the operation of the brain's neocortex, a spiking neural network (SNN) can achieve similar desirable features: flexible learning, speed, and efficiency. Temporal neural networks (TNNs) are SNNs that communicate and process information encoded as relative spike times (in contrast to spike rates). A TNN architecture is proposed, and, as a proof-of-concept, TNN operation is demonstrated within the larger context of online supervised classification. First, through unsupervised learning, a TNN partitions input patterns into clusters based on similarity. The TNN then passes a cluster identifier to a simple online supervised decoder which finishes the classification task. The TNN learning process adjusts synaptic weights by using only signals local to each synapse, and clustering behavior emerges globally. The system architecture is described at an abstraction level analogous to the gate and register transfer levels in conventional digital design. Besides features of the overall architecture, several TNN components are new to this work. Although not addressed directly, the overall research objective is a direct hardware implementation of TNNs. Consequently, all the architecture elements are simple, and processing is done at very low precision. Importantly, low precision leads to very fast learning times. Simulation results using the time-honored MNIST dataset demonstrate learning times at least an order of magnitude faster than other online approaches while providing similar error rates.