These days it is not uncommon to find several so called Olympiads, and that too International. Several are run by private entities, conducted in a few countries, and with the sole objective of making money. These are usually applied through your School. There are certain others that have stood the test of time and grown in stature and popularity.
These are International Olympiads in Physics, Chemistry, Biology, Mathematics, Astronomy, Junior Science, Informatics, Linguistics, Computing, Earth Sciences.
You do not need to apply for these through your school. Anybody who is eligible to appear (mentioned in the rules) can do so.
These exams have multiple levels, where only the top percentage go through, and finally, the best 4-5 students get to represent India at the International Olympiads, competing with about 100 other countries.
For one, it develops your skills in these subjects and helps you determine your passion and your inclination to pursue it in future. Secondly, it demonstrates your excellence and builds confidence. Winning these Olympiads are highly prestigious and are considered the academic equivalents of winning the Olympic Games in Sports and Games.
If you are applying to a University in the United States, winning an Olympiad is a ticket to admission to the very top universities and colleges.