If I've got a patent in a country say India, will it be valid across the world. If not, are there any international patents. If so, who organize them. Is there any facility that, if I file a patent in a country, it has validity in many countries
There is no such thing as an international patent (enforceable rights). To in force a patent you need to go into court. Courts are run by countries who each have their own, differing patent laws.
But there is an international application. The Patent Cooperation Treaty (administered by WIPO) is a way to file one physical application that acts as 140+ virtual applications. PCT applications are filed through your home country or direct with the International Bureau in Geneva. The legal effect is as if you filed an application in all the locations that are PCT members. Most PCT members are countries, but some are patent offices representing a group of countries.
Somewhat separately there are bilateral as well as more complex reciprocal agreements between countries that fall under the umbrella of the Patent Prosecution Highway. PPH agreements allow an applicant with a positive reaction from country A to partition country B to move its counterpart application to the front of the line in country B. There are increasing numbers of ways to get into a PPH program.
In the case of the EPO for Europe. IF the EPO issues a patent it can be turned into a valid enforceable UK patent or Frence or German patent by just paying money and possibly getting official translation of the claims.