The key phrase is "less than" This indicates that we should subtract two things.
... - ...The two terms that are involved in the subtraction are: "15" and "the cube of a number," which means raising to the power of 3. Since "15" comes first in the verbal expression, it should be on the right-hand side of the addition.
a number^3-15
The phrase "a number" is indicating an arbitrary number or a variable. If we call this number x, we can complete our expression.
15 less than the cube of a number
x^3 - 15
Note that this is only one of multiple ways to algebraically express the given verbal expression.