Remove the window from the frame if possible. Newer windows have quick-release tabs on each section. Older double-hung windows are more complicated and best just left in the frame.
Put on leather work gloves then pull out the broken glass window shards. Work carefully since glass shards still have the potential to poke through leather.
Remove the old window putty with a metal putty knife or pry out the wood trim that holds the glass in place with a thin, flat head screwdriver. Save the trim for later and pull out the old nails.
Scrape out the rest of the glass shards and any dirt in the edges of the window frame. Use a sanding block to smooth out the joint, getting into the corners as much as possible. Wipe the surface clean with a damp rag.
Measure the inside opening of the frame with a tape measure, getting an exact measurement up to 1/8-inch. Take this measurement to your local hardware store and get a piece of Plexiglass cut to your exact dimensions.
Run a thin bead of glazing compound along the inside of the window frame and set the Plexiglass in place. Press the acrylic glass slightly to set it into the compound. Use points set around the perimeter to hold the new pane in place.
Replace the window trim, matching each piece up in its original configuration. Hammer the wood in place with 1-inch brads. Use a nailset to sink the ends of the nails below the surface of the wood.
Caulk and paint the edges of the window, without getting paint on the Plexiglass.