Waves in general are energy in motion. There are two types of waves: transverse waves which are like the waves you make if you drop a rock into the water, and mechanical waves, which are most frequently involved with sound. A pulse is a single wave front and is usually shown as moving along a string which has been plucked a single time. A wave is a multitude of pulses, one after the other, that are continuously moving from the point of origin out into space.
SHM, or simple harmonic motion, involves any motion which is regular and symmetrical. A good example is a pendulum which is swinging back and forth. The time it takes to go from left to right and the time it takes to go from right to left are equal, assuming of course there is no friction involved. The period of a pendulum, or any SHM motion is the time it takes to complete one cycle. SHM can also have a frequency, which is the number of complete cycles completed in one second.
The waves you are talking about re X-rays, candles, and sight are all different types of electromagnetic radiation and have different wavelengths, frequencies, and energies. In order to see something, light has to hit the object, reflect off, and enter your eyes where it is converted into electric signals which your brain then interprets as images.
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