A flame sensor is a thin metal rod that sits in the burner flame. Within a couple of seconds of ignition it must detect that a flame is actually burning; if it doesn't, the control board shuts the gas valve as a safety measure so raw gas never pours into your house. It is the furnace's proof-of-flame check.
Over a season the rod accumulates a film of soot and oxidation that insulates it, so it stops 'seeing' the flame even though the burners lit fine. The result is the textbook symptom: the furnace ignites, runs for a few seconds, then shuts off — and repeats, a pattern called short cycling, ending in a no-heat lockout. A dirty flame sensor is the most common cause of that lockout, and one of the most common furnace repair calls overall.
This is the cheapest fix in heating — and the one most worth understanding before you call. The sensor itself is a $10 to $30 part; cleaning it with fine abrasive takes minutes, and a professional cleaning runs about $150 to $200, mostly the service call. A full replacement is still only $75 to $250 installed. Cleaning resolves the large majority of cases; if a sensor needs cleaning more than once in a season it should simply be replaced, because repeated abrasion roughens the metal and speeds corrosion. Be skeptical of any shop that answers a short-cycling furnace with a quote for a control board or a new system before the flame sensor has been cleaned and tested.