Celosia is the main boss of Ghiria. He is guarding the elder of Dinio's village. You were sent to Celosia's chamber, the Flame Temple (this game's Flame Chasm), to rescue the elder, slaying Celosia in the process. He is one of few bosses that is from another game.
To defeat Celosia you need to find a mini tower with a cauldron that has fire in it and climb up it. Celosia will charge at it and cause a stick to fall. This part can be extremely annoying. You must jump off the cauldron and pick up the stick by pressing B. One method used was when he walks backwards, quickly pick up the stick.This will have him come and charge at you. DO NOT panic. Dodge sideways to another cauldron. Climb onto a tower and shake the stick. Get off the tower and approach Celosia with the burning stick; this will scare him. Use this to your advantage and scare him off the cliff (not the one near where you entered, the one on the opposite side of the room, between the columns). This will cause his armor to break. From here you have two options: Jump down onto his back and start attacking, or get on a tower and wait for him to make his way back up. The former saves you time, but the latter is safer. If you choose the latter, shoot an arrow at the colossus when he gets back to the area with the towers, and jump on his back. Either way you do it, be prepared for Celosia to shake violently and run around a lot. The Sigil on Celosia's back is his only weak point.
Just because Celosia is afraid of fire does NOT mean he won't attack you. He will occasionally swipe at you with his claws while you are scaring him to the cliff at the back of the temple, which will knock you back, allowing him to kill you. As well as Celosia still being able to attack you, your flame will eventually go out, so you will want to scare him off the cliff as soon as possible.
Shadow of the Colossus's "Revived Power" plays while figting Celosia in this game. Listen: