The PSTN (Publicly Switched Telephone Network) is the network that carries your voice calls when you call from a landline or cell phone. It refers to the worldwide network of voice-carrying telephone infrastructure, including privately-owned and government-owned infrastrucure.
The PSTN is the network that carries voice calls, and interfacing between the internet and the PSTN has traditionally been very costly and difficult. The Twilio Programmable Voice API abstracts this complexity away, giving developers simple verbs like <Dial> and <Conference> to execute what have been extremely difficult voice calling tasks in the past.