Let $k$ be a field of characteristic not 2. Then a quadratic form $Q$ over a vector space $V$ (over a field $k$ is said to be non-degenerate, if its associated bilinear form: $$B(x,y)=\frac{1}{2}(Q(x+y)-Q(x)-Q(y))$$ is non-degenerate.

If we fix a basis $\boldsymbol{b}$ for $V$ then $Q(x)$ can be written as

$$Q(x)=x^TAx$$

for some symmetric matrix $A$ over $k$ Then it's not hard to see that $Q$ is non-degenerate iff $A$ is non-singular. Because of this, a non-degenerate quadratic form is also known as a non-singular quadratic form. A third name for a non-degenerate quadratic form is that of a regular quadratic form.