Mechanics is one of the most important part of Physics in JEE.
It has huge weightage and not having a command on this part will definitely pull down your rank.
It's important because it has large applications in thermodynamics, electromagnetism as well as in modern physics so being expert in this part helps you to understand the subject better.
Here is a chapterwise analysis of mechanics:
Kinematics: First chapter of mechanics. It's mainly mathematics with a little bit of physics. Knowing calculus is important to understand this chapter. Definitions in this chapter must be absolutely clear and yes focus mainly on projectile motion(especially projectile motion along an inclined plane).
Laws of motion: "backbone of classical physics". Newton's second law is everything here. Practice it well and apply it with caution. Constraint relations form an important part and a number of questions are framed from this topic. Most students face problem in friction part. And yes paper setters know it!
Work, power, energy: work energy theorem and law of conservation of mechanical energy are two important laws that need to be mastered. Don't leave small things out here. In competitions they even ask the units and conversions(for example 1HP = ? Watts).
Center of mass and momentum: another important chapter. The law of conservation of linear momentum is very important and a number of questions are asked from it. Mixed questions from these 4 chapters are common in JEE.
Gravitation: not so important from the point of JEE but very important of you are and olympiad aspirant. If you are done with electrostatics then this chapter is a cakewalk.
Rigid body dynamics/rotational mechanics: most important chapter for JEE mechanics. It was the favourite chapter of paper setters in JEE 2012. Difficult for most students and sometimes confusing too. You are expected to know the formulae, moment of inertia for simple objects, the theorems on inertia, and most importantly - law of conservation of angular momentum. This law combined with law of conservation of linear momentum is used to solve the problem of collisions which is another favourite for competitive exams. Another aspect is impulse momentum theorem and the angular equivalent of the theorem. Brilliant questions have been asked in JEE from this chapter. Even for olympiads, this one is important. Practice this chapter patiently and learn to apply the equations and theorems correctly. Give this chapter extra time.
Simple Harmonic Motion: Another simple yet interesting chapter. Spring mass system is it all about! Make sure you take the help of Work energy theorem to solve the problems of this chapter. All the above chapters are used in solving problems of this chapter.
Fluid mechanics: Pascal's law , Archimedes Principle , Bernoulli's Equation and Equation of continuity are the 4 most important aspects of this chapter. You must be familiar with pressure calculations in different situations. This chapter might prove to be difficult for some (including me).
Elasticity: Easiest chapter of mechanics. Learn the definitions by heart and apply them.
The recommended books have already been mentioned in another post.
Important theorems / laws have been underlined and are expected to be known perfectly.