Twilio Client is a web-based voice client that allows users to make calls using their data connection, bypassing congested public telephone networks. It enables customers and staff to connect to contact centers or business-critical services with reduced congestion issues. Twilio Client can be used for direct routing to existing infrastructure via SIP, routing to non-congested links, and handling security and fraud considerations. The client setup involves generating a valid Access Token, securing the web-based client, and setting up routing options such as direct routing, routing to SIP infrastructure, and routing to an alternate PSTN link.