A Graydle card's effect says when it's destroyed by battle or cards and is sent to the graveyard, it comes back as a equipment card and I take control of an opponent's face up monster. My opponent has a card that says when he destroys a monster and it goes to the graveyard, it comes back to their side of the field. So which effect works first: my card or his? Both say they happen when the Graydle card goes to the graveyard.
It would depend on whose turn it is. The turn player has priority, which means that their effect goes onto the chain first. If the turn player has a Goyo Guardian for example, and they attack a graydle, then Goyo's effect is chain 1. Then the graydle's effect goes onto chain 2. The chain is resolved backwards, which means that the graydle effect happens first, which takes control of the Goyo, and brings back the graydle from the grave, which means that Goyo Guardian's effect would not trigger due to the target no longer being in the grave.
It would be reversed if it was the graydle player's turn. The graydle would get chain 1, and goyo would get chain 2 which, again, resolves backwards. Goyo would revive the graydle, forcing the graydle to be on the field instead of the grave, so the graydle's ability would not be able to trigger.
You can apply the example to cards other than Goyo guardian to get whatever answer you need.