The polarization identity expresses a symmetric bilinear form on a vector space in terms of its associated quadratic form: $ \langle v,w\rangle = \frac{1}{2}(Q(v+w) - Q(v) - Q(w)), $ where $Q(v) = \langle v,v\rangle$. More generally (over fields of characteristic $0$), for any homogeneous polynomial $h(x_1,\dots,x_n)$ of degree $d$ in $n$ variables, there is a unique symmetric $d$-multilinear polynomial $F({\mathbf x}_1,\dots,{\mathbf x}_d)$, where each ${\mathbf x}_i$ consists of $n$ indeterminates, such that $h(x_1,\dots,x_n) = F({\mathbf x},\dots,{\mathbf x})$, where ${\mathbf x} = (x_1,\dots,x_n)$. There is a formula which expresses $F({\mathbf x}_1,\dots,{\mathbf x}_d)$ in terms of $h$, generalizing the above formula for a bilinear form in terms of a quadratic form, and it is also called a polarization identity.
Where did the meaning of "polarization", in this context, come from? Weyl uses it in his book The classical groups (see pp. 5 and 6 on Google books) but I don't know if this is the first place it appeared. Jeff Miller's extensive math etymology website doesn't include this term. See http://jeff560.tripod.com/p.html.