Introduction

Topics

Essentially, olympiads consist of four main subjects: Algebra, Geometry, Number Theory, and Combinatorics. Each of these are broadly independent of the others, and your preparation comes down to your work in each of these four. 

Within each of these subjects, there are various topics: Geometry has topics like Power of a Point and Inversion, and  Combinatorics has topics such as Algorithms and Graph Theory. While learning these topics is important, it is also necessary to note that "knowing the theorem" is not sufficient at all - problems are framed in ways to make you think, and you should have a lot of experience with problem solving within each topic as well.

Computational and Proof based Olympiads

Most countries (including India) begin their math olympiad selection with a "computational" round - it is known as the IOQM/PRMO in India and the AMC, AIME in USA, for example. In these contests, you just need to arrive at the answer by any means, and you don't need to attach any proof. By contrast, most real olympiads (including the IMO itself) require you to submit concrete proofs for each of the problems, with partial markings for progress, and hardly any marks for arriving at an answer without mathematical justification. 

The preparation required for each of the above is slightly different. For the computational contests, it usually just comes down to practice - for which you should find past papers of the contest quite useful. This page will be concerned mostly with the latter, and we'll discuss how we can prepare for proof based olympiads such as RMO/INMO in India or USAJMO/USAMO in USA.

(For computational contests, Ace the AMC 10/12 and Pathfinder to Olympiad Mathematics may be useful.)