Does the british royal family rule over other countries?

The United Kingdom (Wales, Norther Ireland etc.) and the other 15 commonwealth realms.

To help understand this answer a bit better, the Queen is Head of the Commonwealth. In the 20th Century many ex-colonies of the British Empire became independent. Notable among the earliest are Canada, Australia and New Zealand. These countries retained the reigning King, at the time of their independence, as their Head of State, adopting the British system of a Constitutional Monarchy, and a Prime Minister who is head of the government elected to power by the citizens of the country in question. These countries are termed 'Dominions'. All of their parliamentary deliberations are read by the Monarch (currently Queen Elizabeth II), and changes to the law require her signature. She is also responsible for opening and dissolving parliament in these countries, and various other constitutional duties. In practice, however, many of these tasks are delegated to the Governors-General, her representatives in the countries in question. As well as the Dominions, among the 16 countries of which the Monarch is Head of State are the Crown Colonies.

India was the first country to break with precedent and, on independence, became a republic. This has been the pattern for many of the more recently independent nations, all of whom, like India, have a President as Head of State. However, they have chosen to be members of the Commonwealth. This explains why there are 54 nations in the Commonwealth, of which the Queen is Head, but only 16 countries of which she is Sovereign. I hope this rather lengthy explanation clarifies things somewhat! :-)