The Master Sword, Link's mystical darkness-sealing sword, is an optional extra in The Legend of Zelda: Breath of the Wild. You'll want to collect it though, both because of its combat benefits (not only is it unbreakable, it also does double damage within Hyrule Castle grounds) and because of its significance to the game's story.
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The Master Sword is located in the Lost Woods, the entrance to which is a short float away from the Woodland Tower, southwest of Goron Village. You'll want to enter from the ground, though – try to float in and you'll be smothered by fog.
Enter from the path on your map. Don't worry too much – despite the foreboding appearance of the woods, you're not going to encounter any enemies.
Follow the torches, and make sure you don't spend too much time away from them – that fog will pull you right back to the entrance if you stray.
The Lost Woods are huge, but you don't actually need to travel very far to reach the Korok Forest, where the Master Sword is being kept. Set a marker on your map for where the color of the foliage lightens. Torches can be scarce along the way, but if you sprint you should be fine.
You'll find the Deku Tree here. He's been tasked with guarding the Master Sword for the last 100 years. Run inside the Deku Tree for some stores and a place to sleep, or head to your right to find the Keo Ruug Shrine, which as always can be activated as a warp point.
If you've previously met Hestu outside Kakariko Village (he's on the path leading into the village from the south) and offered him korok seeds, this is where he'll set up shop after leaving. Now that you have a shrine warp, you can come back here to keep expanding your inventory slots as you collect more seeds.
Just finding the Master Sword isn't enough, however – you need to be strong enough to pull it out. The Deku Tree is vague on the specifics, but in fact all you need are 13 hearts to retrieve your sword. If you haven't beaten the Divine Beasts nor completed many Shrines, you may need to return later to pull the sword out, King Arthur style. However, if your issue is that you've focused on stamina over health, there's a way to swap your upgrades.
Head to Hateno Village, and look for the Firly Pond on your map (it's at the outer edge of the village, past the model homes). There's a devilish version of a prayer statue here, which will buy upgrades from you and then sell them back at a very lightly inflated price, effectively meaning that you can swap stamina for hearts and vice versa.
Once you have 13 hearts, you will have no problem pulling the sword from the stone, and the final battle will be just a tad easier.