What does the meaning of the autocorrelation in this picture? 
From this figure, how should I understand what is the lag on the top figure? and when in the bottom figure for example the autocorrelation is 0.45 what does tell us about the above figure?
Another small example could be A=[0,1,2,3,4], then the auto-correlation figure is 
So the first one is 1, because the correlation of a vector with itself is 1, then the second number is the correlation between [0,1,2,3] with [1,2,3,4], the second one is the correlation between [0,1,2] with [2,3,4], the third one is the correlation between [0,1] with [3,4].
But when I use Pearson correlation in python I get this numbers?
pearsonr([0,1,2,3],[1,2,3,4])
Out: (1.0, 0.0)

pearsonr([0,1,2],[2,3,4])
Out: (0.9999999999999998, 1.3415758552508151e-08)

pearsonr([0,1],[3,4])

Out: (1.0, 1.0)

How this numbers are associated to the figure 2?
 A: Auto-correlation means how a series correlates with a delayed copy of itself. Lag is the timesteps you move a copy of the series behind to compare against. For example, consider a time series xT where T is the time step, then a lag of 1 will be:
x0 x1 x2 x3
-> x0 x1 x2 x3

So now when you correlate the two series, you look at the span where they overlap:
   x1 x2 x3
   x0 x1 x2

And see how well they vary with each other. Then, for example, a series with lag 0 will have perfect correlation (because lag 0 means the same series). Similarly, a periodic series (for e.g. a sine wave) with a lag equal to its period will have a perfect correlation.
A correlation of 1 means positive linear correlation, -1 means negative linear correlation, and 0 means no linear correlation. For e.g. consider a series 1,2,3,4,5. When you autocorrelate it with a lag of 1, you are comparing 1,2,3,4 with 2,3,4,5. You can see it is a linear correlation. That is, if you plot 1,2,3,4 vs 2,3,4,5, it'll be a straight line.
