A headless server is a computing device without a local interface that is dedicated to providing services to other computers and their users.
Headless, in this context, basically means that the computing device has no monitor or peripherals, such as a keyboard and mouse. Depending on the type of device they run in, headless systems may largely run autonomously or be remotely controlled.
Servers in large data center environments are often headless. An enterprise data center might have hundreds of rack-mounted servers, each of which may be the size of a pizza box. Pizza box servers are rectangular and horizontally-arranged chassis, usually installed in a rack with similar servers.
Tour a Google data center with street view to see some headless servers at work: