In a preprint the author uses the term m-way in the context of tensors without defining it. It cannot be the tensor dimension as the whole sentence says:
"Let $\mathcal{A}$ represent an $m$-way, $n$-dimension symmetric tensor."
What is meant by $m$-way?
Summarizing the commentaries, $m$-way means that the tensor carries $m$ indices $i_1,\ldots i_m$ each of which is from the set $\{1,\ldots,n\}$.
That the author in the mentioned paper calls $n$ a dimension is a little bit awkward.
Rather, I would call $m$ a dimension and $n$ I would call the number of elements per dimension.