Hylian Shield can be found in the Hyrule Castle, the final dungeon of the game. The shield is hidden well. Moreover, getting to it's location will require fighting hordes of enemies and breaking through a real maze of chambers. In the end you will reach the location showed above - use a bomb to break the wall. You will reach a big chamber with bones. When you move a couple steps forward, you will be attacked by a boss - Stalnox.
The fight with Stalnox
This enemy isn't terribly challenging and the tactics required for beating him are rather simple/ You only have to shoot his eye with fire arrows - such attacks will stun him, which will give you a time window when the boss is totally vulnerable. Use this moment to close in and attack with a melee weapon. When the enemy comes back to senses, increase the distance and repeat the procedure. After a couple rounds, the boss will be defeated, and the path to the shield will be open.
After the fight, you can take the shield from the chest.
After you win, you will be able to retrieve a couple items from the boss and pick up the Hylian Shield from a chest - it's the most powerful shield in the game. Along with the Master Sword they combine the iconic equipment of Link and they make the game a whole lot easier.