Read more about Fear from the works of the Mother and Sri Aurobindo.
What is Fear
Fear, desire and sorrow are diseases of the mind; born of its sense of division and limitation.  Fear is a phenomenon of unconsciousness. It is a kind of anguish that comes from ignorance. One does not know the nature of a certain thing, does not know its effect or what will happen, does not know the consequences of one's acts, one does not know so many things; and this ignorance brings fear. One fears what one does not know.  Fear is also a terribly contagious collective thing—it is much more contagious than the most contagious of illnesses. 
Fear is manifested in different parts of selves i.e. physical fear, mental fear and vital fear. In majority of the people, physical is governed by the subconscious and fears that are driven out of active consciousness take refuge in that subconscious. Thus, the physical fear is in the subconscious hidden by the mental and vital fear which are much more conscious. 
Why there is Fear
Fear stems from a sense of inferiority. 
There are three reasons. First, an excessive concern about one's security. Next, what one does not know always gives an uneasy feeling which is translated in the consciousness by fear. And above all, one doesn't have the habit of a spontaneous trust in the Divine. 
One needs to put sincere effort and discipline in overcoming physical fear. To put effort one needs to be courageous in not thinking about the object of fear, because when one fears and constantly imagines the object of fear, then it is as though one is attracting the object of fear. Instead one must use one’s mind to imagine how would one like to respond to the object of fear. Through expanding one's consciousness, by educating it and opening to the divine light one can let go of all grief and fear. If one truly believes in the Divine, surrenders to the Divine, and opens one’s self to its protection and light, putting away one's own ego, fear is bound to go away.  When the fear is deeply rooted, one needs to grip it hard and pull it out, which hurts the surface nature, but if one can have the courage and focus on one’s light, it is possible. One needs to be vigilant and not allow it to come back.  Progress can be really slow but through courage and faith one can fight the adverse forces and expand the consciousness. When one lacks the trust in the divine, they solely depend on themselves. This path can be difficult as one’s trust still lies in ego’s personal independence and there is a refusal to surrender.  Mind and vital are mixed up on the surface of the consciousness, but they are quite separate forces in themselves. As one is able to separate them, their distinct action can be understood and used in overcoming fear. Mind can accept the Divine more easily while the vital is unconvinced and obstinately goes on in search of desire, passion and attraction to the ordinary life. Then, mind under the influence of the soul can be used to govern the vital.
Fear and Illness
Nine-tenths of the danger in an illness comes from fear. Fear can give the apparent symptoms of an illness; and it can give the illness too,—its effects can go so far as that.  Refusing the illness,completely trusting the Divine while infusing oneself with a confidence in the Divine Grace and busying yourself with something else can completely eliminate the fear in the cells of the body and help one not catch even contagious diseases. Fear and Death
One of the most subtle fear is the fear of death which comes from the concern for self-preservation so as to ensure the continuity of consciousness, fearing the unknown. THowever, this fear of death is in the physical cells selves which is beyond reason and almost unconscious. Thus, it is the most difficult to uproot.
This fear goes against the fundamental notion that life is One and immortal. The forms are countless, fleeting and brittle. This understanding can help us overcome fear of death. Thus reason can be used, another method is that of inner seeking i.e. searching the immortal light of psychic consciousness which is in all and identifying with it. Complete surrender to the Divine, and believing that all of us belong to Him integrally. 
The fear of death is not just restricted to the death of the physical, but also the death of the ego, the death of a past version of one’s self, the fear of letting go. When one does not have the courage to live, it is often because something within oneself has to die for something new to be born. It is then one should have that courage, the true courage to abolish the ego. 
Fear and Religion
The fear of the Gods is the beginning of religion.  Fear of the gods arose from man's ignorance of God and his ignorance of the laws that govern the world. The idea of the divine in the form of judge, king exaggerates the importance of the sense of sin and thereby prolongs and increases the soul's fear and self-distrust and weakness. The fear of god is useful only when the soul has not yet grown sufficiently to follow good for its own sake and needs an authority above it whose judgment it can fear and found upon that fear its fidelity to virtue.
Courage is the total absence of fear in any form.  True courage, in its deepest sense, is to be able to face everything, everything in life, from the smallest things to the greatest, from material things to things of the spirit, without a shudder, without physically ….without the heart beginning to beat faster, without the nerves trembling, without the slightest emotion in any part of the being. Face everything with a constant awareness of the Divine Presence, a total self-giving to the Divine, and the whole being unified in this will; then you can go forward in life and face anything whatever.  When the vital is entirely surrendered to the divine it is full of courage, enthusiasm. 
Everyone possesses two opposite tendencies of character, in almost equal proportions, which are like the light and the shadow of the same thing.  Thus, when one has fear, one also has courage. And to get over this impurity of fear one has the resources in the form of courage, tapas and surrender.
Fear is a hidden consent. When one is afraid of something, it means that you admit its possibility and thus strengthen its hand. It can be said that it is a subconscient consent. Fear can be overcome in many ways. The ways of courage, reason, faith, knowledge, surrender are some of them.