Put on some heavy gloves that will protect you from the sharp edges of the broken glass.
Carefully remove the glass from the window. Some will pull out very easily, but others will take some persuading. Use a heat gun to soften up the old glazing compound, and those last pieces will come out with no trouble.
Once the broken glass is removed, prepare to re-glaze it. Use a small wire brush to clean off all the residue on the window frame. If the window is old, apply a bead of linseed oil with a brush. That will condition the wood to bond with the glazing compound.
You'll probably have to warm up the glazing compound by kneading it until you can make a pencil-thin bead out of it. It takes a little practice to get the consistency just right, so don't worry if it's not perfect the first try.
Measure the area to be glazed, write down those measurements and take them to the hardware store. When you order the glass, you need to be as precise as possible with the dimensions.
At the hardware store, pick up a package of push points -- little triangles that will hold the pane in the frame. You'll just push those points right into the sides of the muntin bars. For a small pane of glass, use two points on each side of the glass around all four edges of the pane.
Hold the putty knife at an angle and smooth a bead of putty out in one motion to get an even line. Once it's dry, you can paint it.
To keep that glass crystal clear, don't clean it when the sun is shining on the window. The sun will cause the cleaner to dry too quickly, leaving streaks.