What is the difference between a hot-water heating system and a warm-air heating system?

A hot water heating system (hydronic) uses heated water to circulate from a boiler to baseboard radiators or convectors, where the heat is wanted. This is done through a series of piping and pump(s) that move the heated water around.

Warm air systems use a furnace to heat air and a fan to move the warm air through ducts to the point of use.