As an object following SHM moves, it repeatedly crosses some "zero point" where its amplitude is zero.  For a kid on a swing, this would be when the swing's chain is vertical.  For a mass on a spring, this would be where the mass is/was before it was disturbed.
At the maximum displacement, velocity momentarily becomes zero as the object changes directions (like when you're at the top of your swing).  At these points of maximum displacement, the acceleration of the objects back toward the zero point is the greatest.

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x= X_o cos 2pt/T
a=-a_o cos 2pt/T
v=-v_o sin 2pt/T
Where X_o, -a_{o, and} -v_o are the initial values, t is the time in question, and T is the period.
NOTE: When solving these equations, put your calculator in radian mode!

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DERIVING THE MATH OF SHM
SEE THE RELATION OF CIRCULAR MOTION TO SHM