The normal curve, otherwise known as the bell curve or Gaussian, is a mathematical model. If you want the specifics it is below, but this equation is not necessary to understand the rest, just skip to the point by point reply.
$$ f(x) = ae^{-(x-b)^2 / 2c^2} $$
a is a scalar multiplier, b is the central position and c is the spread or standard deviation. It has the property that the result increases faster and faster until it reaches a central value then declines slower and slower. It is unbounded and the symmetric.
In statistics the Gaussian is standardised to give an integral to 1 (since total probability is 1). This gives gives us a probability distribution curve. Since a Gaussian is symmetric the central position b is the mean of the distribution.
So to your points
- where its mentiond that a population, if plotted against frequency on a bar graph can produce a normal curve (e.g popcorn popping, heights of people etc)
Strictly speaking this is not true which is the whole point of statistics. Real data can never give a true normal distribution. Measurement errors, finite sampling, skews, machine precision, bounds and many more mean we will never get a perfect normal distribution. However, we can compare the distribution obtained to what we would expect to occur under an ideal normal distribution and determine how closely the data matches (this is called a goodness-of-fit test). This allows us to quantify how well the observed data matches or deviates from the normal distribution. There is a popular saying that 'all models are wrong. some are useful' and this is very much the case with the normal curve - it is wrong for real data but it if very often still close enough to be useful.
- Whereas somewhere normal curve is associated with PDF, like its a function where you put infinite random continous variables and plot its probabilities to get a normal curve
Yes. If you want to understand how this works, see above discussion about how Gaussian is converted to PDF.
- Whereas somewhere its written that - use normal curve to take out out inferences out of sample data by using area under the curve and p values and z scores
The normal curve is indeed used in making inferences out of the data. This is because the normal curve is a very nice mathematical model with nice properties, but this does lead to an important point that is critical to understanding how to use statistics and to interpret them.
My original version of this answer was heavily influenced by goodness of fit but without explicitly doing so which made the answer misleading. Statistical tests can be considered a mathematical transformation of multiple samples in a dataset into an overall summary that gives us insight into the overall properties of the data.
The normality assumption is relevant at the point where the statistic is converted into probability. If the distribution of the test statistic is close to a normal distribution then we can use the normal curve to make simplifications in our calculation of that probability. The normal curve is smooth and
differentiable making it very useful for integrating probabilities. We know the raw test statistic includes noise, so if the normal distribution is a good fit for that statistic then projecting the idealised curve onto the data allows smoothing of our probability estimation. The smoothness means that how the p-value varies against the test statistic is consistent and can easily be interpolated for other points. We can easily calculate what values of the statistic are associated with 95 % or 65 % probabilities by simple scaling.
The probabilities of the raw statistic in contrast will jump around with the noise in the data and not be as smooth. Because the raw statistic is not smooth we can't just rescale the value at 65% probability to calculate what the value will be at 95% probability, we would need to recalculate. So we project the model of normality onto the data and use that projection to infer what is occurring in the data.
Note that it is very clear that the relevance of the normality assumption is critical to whether the resulting probabilities are meaningful. Good practice includes verifying if the normal distribution is sufficiently close to the observed statistic distribution for inference to be reliable and meaningful. When we use assumptions of what the distribution of the statistic will look like this is called 'parametric' because we are assuming some parameters apply that can simplify the analysis.
- Whereas somewhere its written that Binomial distribution follow normal curve when number os trials are huge in number
is a special case of
- Whereas somewhere normal curve is assosciated with central limit theorem
The central limit theorem is covered extensively in on CV and it is stating that if independent trials are repeated many times on data then estimates of the mean derived from the replicated trials will tend towards a normal distribution for many distribution types. As you requested no jargon, so I leave the details of what impacts relevance in whuber's comments, suffice to say there are many caveats are there are many examples where it is not known to be applicable. This assumes that the errors are independent between trials, for many cases the CLT collapses if there is any correlation between trials since this can introduce a biased sampling that draws the means towards values that recur across trials. CLT should not be interpreted that any distribution can be fitted by a normal curve if numbers are large enough.
Thus is it used for descriptive stats, or inferential stats?
Remember that the normal curve is simply a mathematical model, it is completely use-agnostic. It is therefore not incompatible that it has relevance for each, but it has a different relevance.
Descriptive statistics don't directly depend on distributional assumptions, these are mathematical operations that convert the variation in the data into a summary value. However, whether a particular descriptive statistic is the most appropriate for insight may depend on the distribution of the data. For example income data tends to be highly skewed by enormous income for the rich and a lower bound of 0 for poor. In these cases the median is often preferred for conveying 'typical' income.
For inferential tests, the data needs to be close enough to normal distribution for test that assume normality to be applicable.
