Specifically, the definition at http://en.wikipedia.org/wiki/Barycentric_coordinates_%28mathematics%29#Definition
Let $x_1, \dots , x_n$ be the vertices of a simplex in a vector space $A$. If, for some point in $A$, $$(a_1 + \dots + a_n)P = (a_1)(x_1) + \dots + (a_n)(x_n)$$ and at least one of $a_1, \dots , a_n$ does not vanish then we say that the coefficients...
Before I finish the definition, I need help in understanding what that first bit means. For example, what does it mean for vertices to be added together and multiplied by another vertex? Or multiplied together? What does it mean to say a point does not vanish?