If R is a regular language, is $R^3= R \circ R \circ R$ also regular? My understanding of a regular language is that for a language to be formal, it must be able to be represented by a DFA or NFA. To prove a language is not regular you can use the pumping lemma to get a contradiction.

I am trying to figure out whether $R^3$ or $R \circ R \circ R$ is regular if R is also regular.

My understanding of the definition of $R \circ R$ is that it is kind of a transitive mapping of R?

And I think if $R \circ R$ is regular then $R \circ R \circ R$ is also regular, so I need to prove $R \circ R$ is regular which really means that I am trying to prove that regular languages are transitive?

I am not sure how I would go about doing this as I am a little confused how regular languages and transitivity relate - as I have only worked with relations for transitivity.
'$R$' 
 is not used in the literature to indicate languages ​​... generally we use $L$.
Anyway, the "$\circ$" operator indicates the concatenation of two or more languages. Keep in mind that the concatenation of regular languages is also regular: you can proove this by 'linking' the NFA's/DFA's thath recognize them.
In this case you are concatenating a regular language with itself, so the resulting language is also regular.

NB: 
The opposite is not always true: if the concatenation of two languages ​​is regular, this does not mean that the languages ​​themselves are necessarily regular.