Badly ripped window screens should be replaced, but most small tears can be easily repaired.
Small Hole Repair
Fill small holes in nylon or fibreglass screens with a few drops of instant adhesive.
Fill small holes in metal screens with epoxy.
Patch kits are often available at home supply and hardware stores; follow the kit instructions to repair the tear.
Patching a Hole
To make your own patch for nylon or fibreglass screens, cut a patch just barely larger than the hole. Apply a thin layer of fast-drying glue along the edges of the patch, and press it into place. To avoid sticking fingers to glue, use a low-tack painter's tape to hold the screen together as it dries; cut a piece of tape larger than the patch, gently tape it to the patch, and leave until dry. To make your own patch for a metal screen, cut a patch from a length of screening. Trim edges of the hole into a neat square opening. Make sure the patch is 1/2 inch larger around than the hole to be repaired; bend edges' teeth into right angles. Set patch over the opening so teeth penetrate the screen. Turn screen over; bend teeth flat on other side to hold in place.
Replacing a Molding Frame Screen
The screening is typically held in place with staples, which are hidden by molding. Pry up the existing molding with a small chisel or screwdriver, and remove the old screening. Cut replacement screening with shears to overlap frame by half an inch all the way around, and set in place. Use staple gun to fasten screen, working from the middle of the opposite sides to the corners, keeping tension even.
Nail the molding
Nail the molding back into place with small nails or brads, and countersink. Trim excess screening with a utility knife. Fill nail holes with paintable wood filler, and paint.
Replacing a channel frame screen
Channel frames come in metal and wood versions. Most modern screens are held in place with a somewhat stiff plastic string called spline, which comes in various widths and styles. Use a small screwdriver to pry up the old spline, and take a piece of this with you to the hardware store for comparison.
Step 1: Cut replacement screening
Cut replacement screening with shears, and position so that it overlaps the channel. "Prime" the screening, unless it is synthetic, by pressing it into the channel along the top of the frame using the convex end of a spline roller, a tool similar to a blunt pizza cutter. This forms a trough for the spline. If you're using synthetic screening, which doesn't hold a shape, simply proceed to step three.
Step 2: Fitting replacement screen
Cut a piece of spline the length of the top edge of the frame. Starting at one corner, carefully push the spline into this channel using the concave side of the spline roller. The fit should be very tight. Repeat this step along the two sides, keeping the screen taut. If a wrinkle appears, pull out the spline, and start over. Finish with the bottom edge.
Step 3: Finish the screen
Use a utility knife to neatly slice away excess screening, being very careful to keep your fingers out of the path of the blade.
Photos from Matthew Septimus and article from Martha Stewart https://www.marthastewart.com/275546/how-to-repair-window-screens re-posted by Pacesetter Homes Regina