The normal distribution is one which appears in a variety of statistical applications. One reason for this is the central limit theorem. This theorem tells us that sums of random variables are approximately normally distributed if the number of observations is large. Normal distribution is one of the general and very common topics which student learn in their course work. Many application based problems comes when we talk about statistics exams and statistics assignments & homework.

Here are the characteristics and properties if normal curves and the normal distribution:
The normal curve is well shaped and symmetrical in its appearance. If the curves were folded along its vertical axis, the two halves would go inside. The number of cases below the mean in a normal distribution is equal to the number of cases above the mean, which make the mean and median coincide. The height of the curve for a positive deviation of three units is the same as the height of the curve for negative deviation of three units.
The height of the normal curve is at its maximum at the mean. Hence the mean and mode of normal distribution coincide. Thus for a normal distribution mean, median and mode are all equal.

There is one maximum point of the curve which occurs at the mean. The height of the curve declines as we go in either direction from the mean. The curve approaches nearer and nearer to the base but it never touched it i.e. the curve is asymptotic to the base on either side. Hence its range is unlimited and in finite in both directions.
Since there is only one maximum point, the normal curve is unimodal.

The point of inflection that is the points where the change in the curvature occurs are X ± s.
As distinguished from binomial and poisson distribution where the variable is discrete, the variable distributes according the normal curve assuming it is a continuous.
The first and third quartiles are equidistant from the median. The mean deviation is forth or more precisely 0.7979 of the standard deviation.
The mean deviation of 4th or more precisely 0.7979 of the standard deviation.
The area under the normal curve distributed as follows:
Mean ±1 s covers 68.27% area; 34.135% area will lie on either side of the mean.
Mean ±2 s covers 95.45% area.
Mean ±3 s covers 99.73% area.