The function of a router in a local area network (LAN) is to direct data traffic between computers and devices, facilitating communication within the network and connecting it to the internet. A router manages data packets by determining the optimal path for their transfer, ensuring efficient and secure communication. It also provides essential functions such as network address translation (NAT) and firewall protection, enhancing security within the network. By enabling multiple devices to share a single internet connection, routers are crucial for both home and business networking environments, enabling seamless communication and data exchange.