The magician asks three volunteers from the audience for rings (friendship, wedding, promise, engagement, mood, toe, etc.) The rings are placed on a pencil (lead, graphite, artists, colored, mechanical, etc.) to prove that the holes in their centers are real. The magician then speaks a magic phrase (perhaps from Lord of the Rings, The Ring, or something of Wagner's) and pours the rings from pencil into hand. With a flourish, the magician displays the rings showing that they are now completely and permanently linked together!
To prove the miracle, the magician shows the linked rings to each volunteer in turn, confirming that their ring is part of the newly made chain. The linked rings are finally presented to one of the volunteers as an amazing souvenir. (Instead of giving the linked rings away, some magicians go through an unlinking process, but this unnecessarily lengthens a routine that has already come to its climax.)
The secret is all in the selection of the rings. When looking for volunteers, try and find a woman who looks like she is engaged to be married, but who is with a man who appears incredibly cheap. With luck, she will be wearing an engagement ring of the vending machine, drugstore, or Crackerjack variety. These rings have a natural break in them to allow easy resizing. Failing this, find a person with a simple wedding band made of soft metal (the purer the gold, for example, the softer).
Once the rings are collected, the magician places them on a pencil (or, in an extreme emergency, a pen). In the process of doing so, if a ring with a break was not found, the magician subtly drops the soft gold ring into a side jacket pocket where a concealed pair of tin snips make the ring more ready to "make magic." If you are worried about destroying someone's wedding band, you have three reasons not to: 1) wedding bands can easily be melted and recast, 2) you may be forgiven if you make sure that the band's owner gets the neat souvenir at the effect's end, and 3) most marriages don't last anyway. (But just to be safe, before performing this effect, make sure you have complete, comprehensive, no-fault liability insurance.)
With the rings on the pencil, it's a simple matter of delicate slight of hand to link the unbroken rings through the gap in the cheap/ruined ring. While speaking the magic phrase, a carefully placed bit of flash paper (impregnated, as magicians know, with thermite for an instantaneous burn temperature of more than 1,000 degrees -- see note about insurance above) is ignited to instantly seal the open ring, forming a perfect chain.
And that's it! After a suitable period of cooling time, the linked rings are given away in a storm of applause.