The CurrentMember function is, probably, the most important function in MDX: understanding how it works is the key to being able to write all kinds of really useful calculations in MDX, such as time intelligence calculations. What it does is this: when you call it from inside a calculation, it tells you where the calculation is being executed inside the space of the cube.
A calculation is executed inside a cell, and a cell has a tuple that points to it; the CurrentMember function allows you to inspect that tuple and find which member from any hierarchy on any dimension in the cube is present in that tuple. This means that the CurrentMember function can only be used with a hierarchy object, and it returns a member object. Using this function is essential while building calculations which include related metrics, such as sales values from different time periods.
If you would like to learn about this function fully, make sure to watch Chris’s in-depth tutorial: MDX Numeric Functions and CURRENTMEMBER.