I'm going through P.J.Olver's book on classical invariant theory. He defines the action of $GL(2)$ on binary forms $Q(x, y)$ by a change of variables:
$$ \bar{x} = \alpha x + \beta y, \ \bar{y} = \gamma x + \delta y. $$
The binary form $Q(x, y)$ is then mapped to a binary form $\bar{Q}(\bar{x}, \bar{y})$ such that the change of variables recovers the original form, i.e. $Q(x, y) = \bar{Q}(\alpha x + \beta y,\gamma x + \delta y) = \bar{Q}(\bar{x}, \bar{y})$.
Later in Chapter 2, he defines that a form $Q(x, y)$ has weight $m$ if the action of $GL(2)$ on a form also incurs the following determinantal factor:
$$ Q(x, y) = (\alpha \delta - \beta \gamma)^m \bar{Q}(\bar{x}, \bar{y}). $$
This means that previously we only considered forms of weight 0. My question is, what are these forms of nonzero weight? Does that mean that I e.g. take a form $a_0 x^2 + 2 a_1 xy + a_2 y^2$ and just redefine my action of $GL(2)$ so that the transformation acquires the factor $(\alpha \delta - \beta \gamma)^m$? If not, then is there an example of, say, a binary form of weight 2?
Yes, judging by the content of Chapter 4 and the rest of Chapter 2, the multiplication by the determinant is a different representation of $GL(2)$ on the space of homogeneous polynomials, called a multiplier representation.