Based upon the banding in the fanned tail feathers, the hawk appears to be in the Accipiter genus, either a Sharp-shinned Hawk or a Cooper's Hawk. Accipiters primarily munch on other birds. These hawks have short wings and a long tail that acts like a rudder, allowing these birds of prey to make abrupt changes in direction as they blast through thick cover, chasing down smaller birds. I've seen these hawks, especially the Sharpies, dive into bushes that seem impossible to fly through. Cooper's Hawks make amazingly quick, acrobatic moves as they chase their prey through trees and yards.
In our urban yards and parks, Cooper's Hawks are more common than Sharpies year-round. But during a spring or fall day of observing hawks at a migration site, like along the lake shore, Sharpies typically far outnumber Cooper's.
The Red-tailed Hawks you see along the roads are in the Buteo genus, and these hawks have long, broad wings and a short tail, and they feed more in open areas on relatively slower-moving prey because these hawks lack the quick maneuverability features of the accipiters.
I'm guessing that KraZyKat's Accipiter was hot on the trail of a bird, and the smaller fleeing bird, say a House Sparrow or a Mourning Dove, bounced off the screen while the hawk failed to react quick enough and went through the screen. Obviously, the hawk would have been moving at a high rate of speed, which they do when they make their surprise attacks. Did you find feathers on the ground near the screen or on the porch?
When a fast-moving Accipiter surprises the typical backyard birds with a sneak attack, the fleeing birds explode in all directions, and they give no thought to where they are going, and that's when they will sometimes hit a window or some other object. Or in this case, they luckily encountered a screen.
I don't think I've heard this kind of hawk story before, so I'll definitely be passing it on. Thanks for sharing.