The headset refers to the system of upper and lower bearings that hold the fork and stem on a bicycle frame.
Currently the industry uses two different names 1. headset, and 2. Aheadset to refer to the bearing/stem system. Even though the names are similar they are completely different. A classic headset requires a fork with a threaded steerer tube to thread through the headset. Then a stem with a quill attaches inside the steerer tube. Cane Creek developed the Aheadset system nearly 20 years ago. It allows for a threadless steerer tube to pass through the bearings. Then a stem clamps directly onto the steerer tube. This system removes much excess metal from the system, the biggest drop in weight comes from the removal of the quill. The headset stays tight by a star nut and cap system that are extremely light.
More recently Aheadsets have evolved into integrated headsets. The metal holding the headset is removed and the bearings are placed directly into the frame's headtube. Today a headset normally refers to a set of bearings, upper and lower. All the excess has left in the past 15 years.