Question: I have a time series of values that may be described as normally skewed or distributed. This is collected from varying degrees of positive and negative integers over time. I then inspect the histogram to see the distribution of these integers and sometimes find an extremely long tail in either the positive or negative direction. 

My question is, since I want to identify these tails, are there methods to determine the point at which the distribution becomes 'long-tailed'? As in is there is a way to quantify the thickness of the tail of a distribution?

Answer: 
You might want to check out extreme value theory (EVT) at this link:

<https://www.statisticshowto.com/extreme-value-distribution/>
It sounds like you want to evaluate the skewness and kurtosis over time.

Skewness being the direction set is "skewed" to one direction or another. Tail left or right.

Kurtosis indicating how long the tails are, but not indicating dominant direction. (Kind of suggesting symmetry.)

But to evaluate over time you need to have sub sample sizes, block sizes, on which to evaluate skewness and kurtosis. The blocks can overlap.

The appropriate sub sample size and overlap depend on your application. As do the values at which skewness and kurtosis become significant.