In the Universe, every body why, every particle is in motion. Nothing is at 'rest'. Rest is death. We never say universe is 'dead'. This motion is of different kinds. 'Linear motion', where the path of the particle may be straight or curved. No body moves for a longer time in a straight path. One cannot move always in straight path however he likes to do so. Whenever a turn comes on the way he has to take a curved path. A straight path may be a shortest path but a curved path is a convenient path. The planets around a sun or electrons around a nucleus move in curved. i.e. a circular path. 
          There is another motion apart from circular motion i.e. 'rotatory motion'. The earth not only revolves in a curved path around the sun but rotates around itself while doing so. Simlarly the electrons in an atom while revolving around the nucleus rotates around themselves.
Another motion : 
          A swing moving between two positions to and from is said to be making 'oscillatory motion'. 
          The 'atoms' in a solid vibrating to and from or up and down between two extreme fixed points and the strings of a violine or a sitar or a guitar.
The stretched membranes of a mridangam or a tabala and the drums, the air columns of a sehnai and a flute or the prongs of a tuning fork in a laboratory when they are played will also make oscillatory motions but with a difference. These motions are along a straight line which are regular such that their displacement is directly proportional to the acceleration, acceleration acting towards mean position where as the displacement is away from it. This popular motion in nature is called 'Simple Harmonic Motion' (SHM).
          Simple harmonic motion is also a periodic motion like circular motion, and oscillatory motion but the last two motions are not simple harmonic. 
          Visible water waves, invisible sound waves, light waves and all the electromagnetic waves are associated with 'SHM'. And thus the motion which is more popular in the nature associated with different forms of energy is 'SHM'. 
          The motion of the needle of a sewing machine, and the wagging of the tail of a dog are 'oscillatory' but not 'simple harmonic'.
Why 'Simple'?
          Because, this motion is sponsored by the projection of a particle making circular motion and circle is considered to be the simplest of all the curves.
Because to draw a circle only one parameter i.e. the 'radius' is sufficient. Since the name 'circular' motion is already given to a particle moving on the circumference of circle, its representitive (projection) a point moving 'to and from' or 'up and down' on its diameter is called 'simple motion'.

          P is a particle making circular motion. PN is a normal dropped from P on the diameter AB of the circle. As P is making circular motion, the foot of the perpendicular N is making a regular up and down motion (vibration) along the diameter AB, AB is a straight line just as the paths of vibrations of the particles associated with music, vibrations of atoms in solid and vibrations of particles associated with water waves and sound waves.

Why 'Harmonic'?
          'Harmonic' means 'regularity' and also 'musical effect' as music is the effect of the particles in regular motion. As this vibratory motion along a straight line is regular and associated with the particles giving musical effect through the instruments. (The name of musical instrument 'Harmonium' is derived for the same reason) This motion is called 'Harmonic motion' and putting the words 'simple' and harmonic together the motion is called 'Simple Harmonic Motion' (SHM).
Mathematical representation of SHM
The above (fig.1) is geometrical representation of SHM.
          In the fig.1 'O' is the mean position. A and B are extreme positions of a particle making SHM, ON = y, the displacement OA = a, the maximum displacement called 'amplitude' 'θ' is called 'phase' (which denotes the state of vibration) of N, the particle making SHM.
From the fig     or y  =  a sinθ which represents SHM (equation for SHM).
Graphical representation of SHM
          A graph is plotted between phase   on X - axis and displacement on Y - axis of N, the particle making SHM.

 Since the angular speed   is constant, 2 is also constant, a  -x which is a necessary condition for SHM
Definition of SHM
            If a particle is making a 'to and from' or 'up and down' motion along a straight line between fixed points. Such that its acceleration is directed towards its mean position, and displacement is away from the mean position and accleration is proportional to the displacement then it is said to be executing 'simple haromonic motion'.

Simple pendulum
            Simple pendulum - Small heavy metal bell tied to a thin string (sewing thread) hanging and swinging to and fro from a rigid support orderly, stylishly is an 'icon' in the romance of physics. The concepts of 'gravity', SHM are associated in its simple 'motion'. Which are the micro scopic characteristics of a macro scopic cosmos. Simple pendulum is 'simple' because its bob executes SHM i.e. the acceleration of its bob is directly proportional to its displacement and oppositely directed.
            Infact, simple pendulum is defined as a 'point mass' (realised in the form of a small, heavy metal bob which is symmetric where the mass is concentrated at a point in its centre),

suspend from a mass less, tension less sewing thread (not a coconut rope). The oscillations (swings) of the pendulum should be free from any other forces except acted on by force of gravity. That way also, the pendulum is 'simple'
Physics concepts in a 'nut shell'!
            The motion of the bob of a simple pendulum involves many basic concepts which govern physics. To set the bob of a pendulum into oscillation, it should be raised to the extreme position to a little extent (i.e. amplitude should be small, why?) against 'force of gravity' for which 'work' is to be done. This work done is stored as 'potential energy' in the bob. Once the bob is left from that position, it fells towards the mean position with greater and greater velocity; Thus the potential energy which is maximum at the extreme position will be converted slowly into kinetic energy which will be maximum at its mean position. Because of this kinetic energy the bob swings towards the other extreme position against the force of gravity and there by converting the kinetic energy into potential energy. Thus, as the bob of the pendulum oscillates energy transforms one form to the other and paving the way for the concept of law of conservation of energy.

The motion of the bob of a pendulum is not only periodic and oscillatory, but also simple harmonic.
Simple pendulum makes SHM
                               
If m is the mass of the bob of a simple pendulum of length 'l', mg is the weight (force) acting down wards which is balanced by the tension 'T' in the string, when the pendulum is in the mean position.
          When the bob is in the extreme position (while oscillating), mg is resolved into two components

i. mg cosθ (where θ is the amplitude) which is balanced by the tension 'T' in the string.
ii. The unbalanced componet mg sinθ is the force which makes the bob to move from A to O with an acceleration.

when θ is small, i.e. amplitude is small (then only it is simple pendulum, otherwise it is a 'swing').

(g is acceleration due to gravity which is constant).
         -ve sign indicates that acceleration is acting towards mean position and displacement (x) is acting away from that position i.e. they are oppositely directed.

Since g and l being constant, acceleration is directly proprotional to the displacement. Thus, the motion of the bob of a simple pendulum is simple harmonic.
Seconds pedulum :
       It is a pendulum where the period of oscillation is 2 seconds.