Extrapolation v. Interpolation What is the difference between extrapolation and interpolation, and what is the most precise way of using these terms?
For example, I have seen a statement in a paper using interpolation as:

"The procedure interpolates the shape of the estimated function between the bin points"

A sentence that uses both extrapolation and interpolation is, for example:

The previous step where we extrapolated the interpolated function using the Kernel method to the left and right temperature tails.

Can someone provide a clear and easy way to distinguish them and guide how to use these terms correctly with an example?
 A: To add a visual explanation to this: let's consider a few points that you plan to model. 

They look like they could be described well with a straight line, so you fit a linear regression to them: 

This regression line lets you both interpolate (generate expected values in between your data points) and extrapolate (generate expected values outside the range of your data points). I've highlighted the extrapolation in red and the biggest region of interpolation in blue. To be clear, even the tiny regions between the points are interpolated, but I'm only highlighting the big one here.

Why is extrapolation generally more of a concern? Because you're usually much less certain about the shape of the relationship outside the range of your data. Consider what might happen when you collect a few more data points (hollow circles):

It turns out that the relationship was not captured well with your hypothesized relationship after all. The predictions in the extrapolated region are way off. Even if you had guessed the precise function that describes this nonlinear relationship correctly, your data did not extend over enough of a range for you to capture the nonlinearity well, so you may still have been pretty far off. Note that this is a problem not just for linear regression, but for any relationship at all - this is why extrapolation is considered dangerous. 
Predictions in the interpolated region are also incorrect because of the lack of nonlinearity in the fit, but their prediction error is much lower. There's no guarantee that you won't have an unexpected relationship in between your points (i.e. the region of interpolation), but it's generally less likely. 

I will add that extrapolation is not always a terrible idea - if you extrapolate a tiny bit outside the range of your data, you're probably not going to be very wrong (though it is possible!). Ancients who had no good scientific model of the world would not have been far wrong if they forecast that the sun would rise again the next day and the day after that (though one day far into the future, even this will fail). 
And sometimes, extrapolation can even be informative - for example, simple short-term extrapolations of the exponential increase in atmospheric CO$_2$ have been reasonably accurate over the past few decades. If you were a student who didn't have scientific expertise but wanted a rough, short-term forecast, this would have given you fairly reasonable results. But the farther away from your data you extrapolate, the more likely your prediction is likely to fail, and fail disastrously, as described very nicely in this great thread: What is wrong with extrapolation? (thanks to @J.M.isnotastatistician for reminding me of that). 
Edit based on comments: whether interpolating or extrapolating, it's always best to have some theory to ground expectations. If theory-free modelling must be done, the risk from interpolation is usually less than that from extrapolation. That said, as the gap between data points increases in magnitude, interpolation also becomes more and more fraught with risk. 
A: In essence interpolation is an operation within the data support, or between existing known data points; extrapolation is beyond the data support. Otherwise put, the criterion is: where are the missing values?  
One reason for the distinction is that extrapolation is usually more difficult to do well, and even dangerous, statistically if not practically. That is not always true: for example, river floods may overwhelm the means of measuring discharge or even stage (vertical level), tearing a hole in the measured record. In those circumstances, interpolation of discharge or stage is difficult too and being within the data support does not help much. 
In the long run, qualitative change usually supersedes quantitative change. Around 1900 there was much concern that growth in horse-drawn traffic would swamp cities with mostly unwanted excrement. The exponential in excrement was superseded by the internal combustion engine and its different exponentials. 

A trend is a trend is a trend,
  But the question is, will it bend?
  Will it alter its course
  Through some unforeseen force
  And come to a premature end? 
-- Alexander Cairncross 
Cairncross, A. 1969. Economic forecasting. The Economic Journal, 79: 797-812. doi:10.2307/2229792 (quotation on p.797) 

A: Example:
Study:  Want to fit a simple linear regression on the height on the age for girls of age 6-15 years old. Sample size is 100, age is calculated by (date of measuring - date of birth)/365.25. 
After data collection, model is fit and get the estimate of intercept b0 and slope b1. it means we have E(height|age) = b0 + b1*age.
When you want the mean height for age 13, you find that there is no 13 year old girl in your sample of 100 girls, one of them is 12.83 years old and one is 13.24. 
Now you plug in age = 13 into formula E(height|age) = b0 + b1*age. It is called interpolation because 13 year old is covered by the range of your data used to fit model.
If you want to get mean height for age 30 and use that formula, that is called extrapolation, because age 30 is out of the range of the age covered by your data.
If the model has several covariates, you need to be careful because it is hard to draw the border that data covered.
In statistics, we do not advocate extrapolation.
A: TL;DR version:  


*

*Interpolation takes place between existing data points.

*Extrapolation takes place beyond them.  


Mnemonic: interpolation => inside.
FWIW: The prefix inter- means between, and extra- means beyond.  Think also of interstate highways which go between states, or extraterrestrials from beyond our planet.
A: The extrapolation v.s. interpolation also applys in neural networks as mentioned in Rethinking Eliminative Connectionism and Deep Learning: A Critical Appraisal:

generalization can be thought of as coming in two flavors,
interpolation between known examples, and extrapolation, which
requires going beyond a space of known training examples

The author wrote that extrapolation is a wall stopping us reaching artificial general intelligence.
Let's suppose that we train a translation model to translate English to German very well with tons of data, we can be sure that it can fail a test with randomly permutated English words because it has never seen such data in the training process and it is certain to fail a new phrase coined after it is trained. That is it behaves badly for open-ended inferences because it can be only accurat for data similar to the training ones but the real world is open-ended.
References:

*

*Extrapolation in NLP

*Real Artificial Intelligence: Understanding Extrapolation vs Generalization
