A lot of books and sites have alternate names for this wonderful material, because they want to avoid getting in trouble for using a trademarked name. Well, too bad. If a person knows they want homemade Silly Putty, how are they to find the recipe online if people are calling it things like "Fun Putty?"
Anyway, to make your own Silly Putty, all you need to do is mix together roughly equal parts of white school glue and liquid starch. We get the glue in gallon containers and the starch is easy to find at any grocery store. If you only want a little, say less than a cup, measure out equal parts of each ingredient. You might want to go this route if this is your first time making the stuff. Try starting with a tablespoon of each.
If you want a lot, pour your glue into a big plastic bowl and then add liquid starch a little at a time. Don't bother measuring. Stir and knead until it reaches the right consistency - you'll know it when it happens. It will be smooth and rubbery. I like to just dump in the ingredients and hand the bowl to a child. Play your cards right and you'll never have to touch the stuff (although you'll probably want to!) I've also let each person make their own by pouring the glue and starch onto styrofoam plates. A chemical reaction causes the glue to become stretchy and more solid. You may need to drain off excess starch after mixing. If kids are doing the mixing you can have them gently dab their putty with a paper towel.
I've also made the same material using that blue gel glue. It comes out a transparent blue, which is really neat. It's stiffer than the white glue version, and not really stretchy, so I just kept it in a plastic bag and let the kids play with it that way. The result here might depend on the brand I used, so you should experiment to see what you can get with your locally available types.
You can add in food coloring or liquid watercolor to your Silly Putty for even more fun. Divide up the batch and make several colors for an entertaining color-mixing experiment. I wouldn't let kids do the initial mixing with their hands if you're using food coloring, though, because the concentrated stuff will really stain their skin and clothes. Stir the color into the putty a bit before you give it to them.
The more you play with this stuff, the better it gets, so let the kids smoosh it up for a good long time. You can use this putty the same way "real" Silly Putty is used... Pull it slowly so it stretches, and snap it fast so it breaks. Press it into molds, cut it with scissors or cookie cutters, or try using Play-Doh tools with it. It will also lift images from newsprint. Clean-up is pretty easy as long as you limit the kids to using it in one area. Putting down newspaper helps. Vinegar will take the stuff out of clothes, carpets and hair. When you're done, you can store the putty in Zip-loc bags, margarine tubs, or plastic easter eggs.
Fill up your water table with the stuff. It might give your director a heart attack, but it sure will entertain the kids. Good times.