Another (maybe more intuitive) way to look at affine transformations is to define them as transformations which preserve points, lines and planes. Parallel lines must stay parallel, meaning you can have transformations that turn squares into parallelograms, but not into trapezoids.
To add onto nahman, affine transformation allow us to use homogenous coordinates to carry out transformations such as 2D translation and 3D transformations (e.g. rotation, reflection, etc). An important note is that affine transformation preserve the the ratio of distance between points lying on a straight line and the barycenters of a weighted collection of points.