You will need a screen with artwork on it, ink, squeegee, balloons and a means of blowing them up. And a paper clip or clothes peg to seal the balloons after inflation. Also white spirit for cleaning off the screen, rubber gloves to protect your hands, and paper towels and some old newspaper to put everything on. And brown parcel tape or sellotape. To shrink the balloons down to their original shape you need a hairdryer to provide hot air.
You also need a cardboard box measuring approximately 9 inches by 8 inches by 6 inches to hold the balloon steady for printing. This box has to have one 9 by 8 side removed so that you can lay the inflated balloon in the box. Then I have found that a vertical slit in the cardboard at one end will effectively seal the inflated balloon instead of a clip or peg, see more details below.
If there are any clear areas on the screen apart from the printing image you will need to cover them with tape. Apply this to the underneath of the screen i.e. the side where the image is back to front. You also might want to apply tape around the edges to stop ink leaking through.
Inflate a balloon to about 9 inches diameter or two thirds of its normal size. Twist the neck a couple of turns and apply a clip or peg. If using paper clips be careful not to pierce the rubber. Air may escape slowly but the printing should not take long.
Place the inflated balloon in the open box so that one side of the balloon is upwards and the nozzle is at the bottom. Instead of clips you can slide the neck of the balloon into the vertical slit you made at one end of the box.
Take the screen and dribble a line of ink just above the image, near the top edge of the screen. Take the screen in one hand and the squeegee in the other. Press the screen on top of the balloon and with one or two strokes wipe the squeegee blade over the image, covering it with ink. Hold the squeegee almost vertical, tilted slightly towards you. Pull the squeegee towards you from the top of the screen to the bottom. It is best if you only stroke each part of the image once or you may get a double blurred print.
Once all the image is printed lift the screen vertically upwards to remove it from the balloon. It may stick slightly but ignore this. If you remove the screen smartly without any sideways movement the printed image should be nice and sharp.
The ink on the balloon normally air-dries while the balloon is inflated but with white or coloured ink you might need to give it a blast of hot air from your handy hairdryer!
Put the screen to one side and take the clip off the balloon, check the ink has dried and then you can allow it to deflate, put it in a container and move onto printing the next one.
You may waste a few balloons getting the hang of it. The main factors in successful screen-printing are 1) having a perfect screen to print with 2)using the squeegee rapidly and confidently and not doing lots of small uncertain strokes 3)keeping the ink in its place which means not getting it on the underside of the screen, on the squeegee handle and then on your fingers and everywhere else!
If it all goes wrong clean everything off with white spirit so you can begin again with a clean uninked screen.