It is a golden-bronze ring with SpongeBob and Patrick inside it. The song they sing goes like this:
It's the best friends forever,
Best friends forever, ring!
Role in series
"The Secret Box"
Patrick tells SpongeBob that if he tries to peak into his secret box one more time, they will no longer be friends, in regret of joining the Best Friends Forever Club. The camera then zooms in on the friendship ring on Patrick's finger.
Later at night, when SpongeBob breaks into Patrick's house, Patrick suspects that the clam burglar has broken into his house, but it actually turns out to be SpongeBob, as he reveals his true identity by showing his the friendship ring on his finger.
When SpongeBob is singing "Sweet Victory," he pulls out his hand and the friendship ring is briefly shown, though it's maybe possible that the ring is a part of the band uniform.
SpongeBob's Game Frenzy
I can turn into a skyscraper!
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Coincidentally, both of the episodes that the friendship ring appears in are sister episodes.