Since the sun shines more strongly on the equatorial regions, the air there
would be heating to a greater extent than at the poles.
This hot air would rise, creating an equatorial low pressure system.
This low pressure system would draw in air from the mid-latitudes.
Meanwhile, the air in the atmosphere would be spreading out towards the poles.
This air would be cooling (due to lower pressure as it rises) and would eventually
sink at the poles, creating a high pressure region there.
Finally, this high pressure region would help push air back towards the
equator.
Once started, this convection cell would continue on both sides of the equator.
To explain cause of the "Tradewinds," Edmund Halley first postulated the
existence of convection cells in the earth's atmosphere in 1686. In 1735,
George Hadley extended Halley's explanation by adding the Coriolis effect
(it didn't have that name at that time-- Gaspard de Coriolis later explained the
mathematics behind the effect, and it has been named after him) which explained
the easterly component of the tradewinds.