I explain in this answer what a projective transformation (aka projectivity or homography) is. It's a function $h$ of the form $$h: \mathbb{P}^2 \rightarrow \mathbb{P}^2,$$ where $\mathbb{P}^2$ is a projective space, so, essentially, a 3-dimensional Euclidean space of homogenous vectors.
You can also represent a homography as a $3 \times 3$ matrix $\mathbf{H}$, so that, when we apply this projective transformation to some input $\mathbf{x} \in \mathbb{P}^2$, we get $\mathbf{x}' \in \mathbb{P}^2$, so we can represent a projective transformation as follows.
$$\mathbf{H}\mathbf{x} = \mathbf{x}'$$
So, basically, a projective transformation is a linear transformation between projective spaces.
You can generalize these ideas to higher-dimensional projective spaces, i.e. $\mathbb{P}^n$.
Although you can represent a projective transformation as a matrix multiplication, there's more to it. In fact, it's a linear transformation, with 8 degrees of freedom, between projective spaces. You can also view a homography as a generalization of other transformations, like isometries, similarities, and affinities. This is explained more in detail in chapter 2 of the book Multiple View Geometry in Computer Vision (2nd edition) Richard Hartley and Andrew Zisserman.
