Reproducing ROW_NUMBER() in MySQL

Recently I was working on a project involving MySQL and needed to partition and rank results in a SELECT statement. This is a pretty straightforward approach in SQL Server using ROW_NUMBER() or RANK(). In my particular case I needed functionality equivalent to ROW_NUMBER() with partitioning by a single column and sorting by another. Below is an alternative since earlier versions of MySQL do not have an equivalent ROW_NUMBER() function.