Wet and dry seasons - what are they and how do they form?
In the mid-latitudes we experience 4 seasons (winter, spring, summer and autumn) due to the large annual variation in sunlight due to the tilt of the earth in its orbit.   The tropical latitudes, however, do not see large annual variations in sunlight or temperature and instead define their seasons by precipitation changes e.g. wet and dry seasons. 

Some examples of a wet/dry climate are monsoon climates and savanna climates.

The monsoon is driven by seasonal changes in wind directions.  The typical setup is offshore flow during the dry season and onshore flow in the wet season with a mechanism similar to the sea breeze (e.g. driven by temperature differences between land and sea).  During the dry season, dry  air flows out to sea and during the wet season, moist air flows inland from the sea, rises and produces rain.  The monsoon circulation occurs in parts of Africa, Asia, Northern Australia and arguably in the North America desert southwest (debatable because it lacks a true seasonal flow reversal).  

The savanna climate generally have a dry season of varying intensity and length and a rainy season in the remainder of the year.  Total rainfall in savannas tends to be less than that in monsoon regions.  

Both of these wet/dry climates involve interaction with the inter tropical convergence zone (ITCZ), which is a persistent band of rising air and thunderstorms near the equator.  It moves north and south with changes in location of the sun seasonally and responds to the monsoon circulations and ends up merging with them.   

The other tropical climates are wet all year long (rainforest climates) and dry all year long (desert climates).  The rain forests are equatorial and dominated by the ITCZ year round.  The hot deserts are around 15-30 N/S latitude and are favored on the western extents of the continents.  The latitude preference derives from the descending branch of the Hadley cell, which suppresses precipitation and causes warming.  The western continental preference comes from the direction of the trade winds, which are easterlies in the tropics.