I can't remember ever causing a hole in a screen door or screen on a window. Yet and still, I can't remember ever living anywhere where there were not at least a couple of holes in said wire screen doors or window screens. If you live in a place such as I, and one of your largest fears is having a mosquito eat your dead ass up while you're sleeping on a warm summer night, having holes in any screens designed to protect you is akin to having your arch-nemesis sneak in one day and cut those little tips off of all your condom collection. The common joke around here is that we have mosquitoes so big that they can stand flat-footed and fuck a turkey. In fact, they are not that large. They can easily fly through small holes in a screen. They only seem that large when you wake up and find yourself covered in those gigantic swellings which will itch you nearly to the insane asylum before they calm down and disappear.
There are other nasty bug-creatures which can come in through these holes in the screens, so let's forget about this irrational hatred I have for mosquitoes and think about how to repair those holes.
There are basically three methods for fixing small holes. If the holes are large enough toss your empty beer cans through (in order not to clutter up the living room any further), then you might want to go ahead and bite the bullet and buy a new screen door or window screen. However, if the hole is rather small (such as that 9mm bullet hole where you tried to shoot that squirrel who was eating out of your bird feeder that Saturday afternoon just before the cops arrived), these are the methods for fixing it.
Get some heavy duty thread and a needle and just sew up the hole by tracing the pattern of the mesh, using the nearest viable screen around the hole. The downsides to this method are numerous. The thread will likely not look like the screen, and if you're an anal retentive disaster of a human being, it will annoy you every time you look at it. Also, the process will be very time consuming. And you will have to be very careful not to pull too tightly every time you thread a new loop, or you will pull the viable screen out of place and have to start all over.
Cut a piece of screen about an inch larger that the hole and either use thread or a long piece of one wire from the same screen material to sew around the edges of this new piece of screen when it is in place. Again, the downside is that the process is very time consuming. Also, you're going to need a pretty long piece of wire.
The better way is to cut a piece of screen about an inch and a half larger that the hole and then remove about half an inch of the screen wire on all four sides of your patch. Bend the lonely wires at a 90o angle and push the patch through the screen over the hole. Then, all you have to do is fold over the wires from the other side in order to make a patch that should last a while.