The riddle
What Can Fill a Room but Takes Up No Space?
Why light fits and air does not
Air is a common first guess, and it fails. The International Standard Atmosphere puts air at sea level at 1.225 kilograms per cubic meter, according to Wikipedia. A room 4 meters by 5 meters with a 2.5 meter ceiling holds 50 cubic meters, which is about 61 kilograms (about 135 pounds) of air. That is plenty of material taking up space. Light is not made of anything like that.
What light is made of
Wikipedia describes photons as massless particles that can only move at one speed, the speed of light. They belong to the class of bosons, and the article notes that they do not obey the Pauli exclusion principle, so more than one photon can occupy the same quantum state. The word photon, from the Greek for light, was popularized in 1926 by Gilbert N. Lewis. The riddle's claim is about making room: switching on a lamp does not push the air or anything else out of the way.
Crossing a room
The speed of light in a vacuum is exactly 299,792,458 meters per second, by the definition of the meter. A 5 meter room is crossed in about 17 billionths of a second. Switch on a lamp and the whole room is lit faster than any person can notice.
Other answers
Some tellers answer sound or silence, since a sound can also fill a room without crowding anything. Light is the answer most versions give. For another question in the same style, see what has no beginning, middle or end.