Blow up a balloon to about 2/3 it's maximum size and tie a knot in the end. Place a little oil on the end of the needle and on the far end of the balloon (this allows the needle slide through the balloon easier). Slowly push the needle into the balloon. If the needle is sufficiently sharp and smooth it will not tear the rubber but will slide between the polymer chains, allowing them to stretch around the needle. The needle should then pushed through the balloon until it comes through the other side near the knotted end.
On a molecular level, a polymer is a lot like a long paper clip chain. Each paper clip represents a simple molecule called a monomer ("mono" means one and "meros" means unit). The monomers link together to form a long molecular chain called a polymer ("poly" mean many). Natural rubber is a polymer composed of repeating units of the monomer isoprene:
CH3The polymer chains in rubber are folded and are joined to each other in a network structure that imparts a high degree of flexibility. Magicians take advantage of this characteristic in the needle-through-the balloon-trick.