There are 'Royal Societies' for all sorts of organisations - for example - RSPCA (Royal Society for the Prevention of Cruelty to Animals) or RSPCC (Royal Socety for the Prevention of Cruelty to Children) and many more. It just means that they have royal patronage.
One organization which is often referred to as simply "The Royal Society" is The Royal Society of London for Improving Natural Knowledge.
It was established in 1662, and nowadays acts as an advisory body to the British Government, and as the UK Academy of science as well as funding scientific research.
The Royal Society was founded in November 1660 and was granted a Royal Charter by King Charles II as the Royal Society of London. The Society is a learned society for science.
The Royal Society for the Prevention of Cruelty to Animals.
Royal Society was created in 1660.
Royal Society of Canada was created in 1882.
Royal Watercolour Society was created in 1804.
Royal Agricultural Society was created in 1838.
Royal India Society was created in 1910.
Royal Society of Chemistry was created in 1980.
Royal Historical Society was created in 1868.
Royal Society of Queensland was created in 1884.
Royal Statistical Society was created in 1834.
Royal Dublin Society was created in 1731.