Suppose I have a data set like the image below. The question is which is dimension here rows or columns?
For a clustering algorithm like you suggest in your comments, each row should represent a point. This point is in a space that has some dimensionality characterized by the number of columns. So here you have four points (Row1, Row2, Row3, Row4) in a four-dimensional space. When you read about "dimensionality" in a data science context, it's talking about the number of attributes for each data point, not the number of data points. This is separate from the representation of the data as, for instance, a 2D array.