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Consider this situation given in Principles of Physics:

Push horizontally on a heavy crate. The crate doesn't move. From Newton's second law, a second force must also be acting on the crate to counteract your force. Moreover, this second force must be directed opposite your force and have the same magnitude as your force, so that the two forces balance. That second force is the frictional force.

However, I am not in agreement with this because I feel that "second force" is the normal force exerted by the crate on me which is equal to the force that I exert on the crate (Newton's Third Law).

Could someone please explain how my understanding of this concept of static friction wrong? Also, please try to support your answers with diagrams which would help me visualise the matter more clearly.

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The reaction/normal force that acts on you does not act on the crate. Therefore another force is still required to counteract your pushing, which is the frictional force.

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Try to understand that action-reaction pairs act on different bodies, and hence they never cancel out each other.

You do get a normal force, but that force is exerted on you, so that cannot counterbalance the force you are exerting on the crate. In order to counterbalance the force you exert, a frictional force is acting on the crate. When not moving, this frictional force $$f_s = \mu_s N $$ where $\mu_s $ is the coefficient of static friction.

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    $\begingroup$ The static friction force increases up to the limit given by your equation, but it is not necessarily constant. $\endgroup$
    – David White
    Jul 30, 2017 at 14:39
  • $\begingroup$ I never said it's constant. Did I? $\endgroup$
    – Wrichik Basu
    Jul 30, 2017 at 14:49
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    $\begingroup$ No. However, a clarification of the wording in your last sentence would be helpful. When not moving, the static friction force equals the force on the object, and is directed opposite the applied force, up to the maximum static friction force given in the equation. The equal sign in the equation implies a constant force. $\endgroup$
    – David White
    Jul 30, 2017 at 19:49
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The crate being stationary implies that the net force on the crate is zero. When you push the crate, the crate pushes you back with an equal and opposite force but still the net force on the crate is not zero and the crate would accelerate had the frictional force not been there.

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Remember that the normal force is always perpendicular. You are right in what you say, you just missed the point that friction and normal force take care.of two different directions.

If you push at an angle on something, then a reaction force appears to hold back. The perpendicular part of this force is called the normal force, and the parallel part is called a friction force.

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