what type of regression? I am looking to explore the relationship between two variables and am trying to find most appropriate statistical test. 
My data comprises two variables, one is a count of the words shared between two utterances in conversation, the other is a distance measure between those two utterances (measured in utterances); 
speaker A: hello there how are you?
speaker B: hello I am doing well, you?
speaker C: look over there

For each distance measure, I count the number of words shared. For example, distance = 1 would have a count of 2, because speakers A and B share two words ('hello' and 'you'), and speakers B and C share no words. Similarly, distance = 2 would have a count of 1 because speakers A and C (being two utterances apart) share one word ('there').
I have done this over a corpus of conversations and have a table that now looks similar to the below:
convo  |  distance  |  word_count
---------------------------------
 1           1            10
 1           2            12
 1           3            8
 2           1            17
 2           2            4
 2           3            14

I was hoping to use some sort of regression to be able to say test the relationship between distance and count, but as both are whole numbers, and distance in particular occurs only a small range (up to 5 in my dataset) I'm unsure where to start searching. 
 A: You could try with Poisson regression since your dependent variable is count data i.e. discrete values 1,2,3 etc. and this type of model is meant for this kind of data. You will have to check some assumptions such as equidispersion (equality of mean and variance).
I recommend you to look at:
A. C. Cameron and P. K. Trivedi (2005), "Microeconometrics: Methods and Applications," Cambridge University Press, New York. Chapter 20
Greene, W. H. (2000). Econometric analysis. Upper Saddle River, N.J: Prentice Hall.  Chapter 21
You can have examples on Poisson regression implementations in R here:
https://stats.idre.ucla.edu/r/dae/poisson-regression/
https://www.dataquest.io/blog/tutorial-poisson-regression-in-r/
Just be aware that if your dependent variable, distance, has an excess of zeros such that your distribution is highly skewed, you may want to consider zero-inflated Poisson models or Negative-Binomial models depending on the data and assumptions. More examples here: 
https://stats.idre.ucla.edu/r/dae/zip/
