Vincents answer is pretty complete, but a bit technical...I'll try a less technical more discussion answer.
First to note...we don't fully understand why. These are patterns, we can observe them, but whats actually causing them isn't well known. The ocean heavily influences our climate and it has a few cycles that are multi-year cycles that impacts the wet/dry seasons...El Nino / La Nina are pacific ocean patterns and the El Nino is tied to the Indian monsoons, Caribbean rainfall, and California's dry season (it effects the Saharan dust layer as well...a not well understood pacific ocean cycle has world wide impacts). It's why your question ends up in world building and not Earth Science...we are still in semi-speculative territory with this. As a general pattern...if you don't experience much of a winter/summer seasonal change, odds are you'll see a dry/wet season instead.
If you want the simplest explanation...the wind patterns tend to form a 'rain belt' (defer to Vincents answer and ITCZ explanation). This rain belt moves over an area during it's rainy season and elsewhere for it's dry season. In the case of India, winds from the Southwest pick up rain clouds from the Indian ocean and move them inland creating the monsoons. During the dry season, these winds reverse and they come from the North East as relatively dry winds.
The reasons for this are significantly more complicated...but keeping simplistic: if you take a globe and view it from the North pole, then flip it over and view it from the south pole, you will notice there is a significantly greater amount of land in the Northern Hemisphere vs a much larger percentage of ocean in the southern hemisphere. The oceans heat differently than the land does...oceans almost entirely absorb the suns energy while the land tends to reflect it much more. The ocean sucks up the energy from the sun, but the majority of it goes to changing water from a liquid to a gas and not heat changes...while on land this energy tends to raise the temperature of the air instead. This seasonal variation in the amount of sunlight the Earth absorbs is heavily responsible for the wet/dry season. Using India as an example again...the Northern regions of India sees a high pressure system dominate during the winter which drives the winds from the North East and keeps the region dry. During the summer months, the Indian ocean greatly warms and it shifts to a low pressure system over the Thar desert/Northern and central India with the high pressure system over the Indian ocean. This pushes the majority of the humidity out of the Indian Ocean and over India instead.
You would have to give further descriptions to get much more in detail than this. I'd go with the general rule that any place considered tropical and between the tropic of Cancer and Capricorn would see a dry/wet season instead of a cold/hot season