Evil Eagle is the boss of the seventh dungeon of Zelda: Link's Awakening, the Eagle's Tower.
If you're looking for more assistance, our Zelda: Link's Awakening walkthrough can help.
Evil Eagle boss fight
For this fight, you need to have the Bow (and a healthy supply of arrows) or ideally the Boomerang, which you can get at the end of the trading sequence quest.
With these projectiles, not only can you damage the boss, but halt its many attacks. The bird can whizz across the screen - sometimes at the same height as you - and flap its wings at the side to blow you off the edge, and / or send damaging feathers your way.
But if you launch a projectile at any of these, it will fly away and start the next attack.
If you are caught off guard and cannot attack when it is, say, flapping its wings and attempting to glow you off the side, then you can hold up your shield to block any feathers and advance towards it.
There are spikes below you can fall off onto, though if you move towards the centre again, chances you will miss them.
Once done, grab the Heart Container and down the ladder again. Back in the dungeon, the right door will now open, taking you to a room with the Organ of Evening Calm.
Our Zelda: Link's Awakening walkthrough and guide can help you with the critical path, including the Eagle's Tower and Turtle Rock dungeons. There's plenty to be done on the side, too - including Zelda Link's Awakening Heart Pieces, Zelda Secret Seashells and the Zelda Link's Awakening trading sequence. Finally, there's also the new Zelda Chamber Dungeons feature to unlock and complete.