Question

is the monsoon responsible for rains the world over

Answer 1

No, Monsoon alone is not the reason for rains. Rains occur everywhere. 
But Monsoon is a phenomenon that occurs in some areas of the world. True monsoon region is South East and South Asia, extending till Japan & Korea past Chinese coast. The other monsoon regions are - Islands of Caribbean Sea- Mexican southern coast & Yucatan peninsula; African coast of Gulf of Guinea; Northern Australian (York and Arnheim peninsulas) Coast around the Gulf of Carpentaria. 
Monsoon is the result of shifting of ITCZ (Inter tropical convergence zone) that follows sun from north to south and back. Technically it is defined as (low-pressure) trough of ITCZ. 
Asian monsoon is said to be because of intense summer heating of very high-altitude Tibet plateau creating a vast low pressure area there. It is called Summer monsoon, South West (SW) Monsoon also. This sucks the ocean winds over (West) Pacific, past Indonesian archipelago and (north-east) Indian Ocean. The Topography, Orography (alignment of mountain ranges) like Himalayas, Western Ghats (Indian peninsula), Annam highlands (Vietnam) shape the monsoon and its intensity. The moisture laden ocean winds when hitting the heated land rise and suffer adaibatic expansion and cools. The moisture condenses as a result and torrential rain results. in the hills north of Bangladesh in India (Khasi-Garo hills) rainfall may reach 10,000mm in that 4 month period. The scorching Indian summer climaxes with the monsoon clouds building up from eastern sky like a herd of elephants with lot of thunder and lightning and drenching of the land. After this, suddenly the monsoon reverses and blows in opposite (from NE) direction. While retracing its path over Bay of Bengal, the moisture laden winds hit South-Eastern Indian peninsula (Tamil Nadu) where it is the main monsoon occuring in October-December.

Answer 2

No monsoon alone is not responsible for rains .Rains occur everywhere.
But Monsoon is a phenomenon that occurs in occurs in some areas of the world.True monsoon region is South East and South asia, extending till Japan and Korea past Chinese post.The other regions are Islands of CaribbeanSea and many more.