How do you beat a Stone Talus in Legend of Zelda BoTW?

In order to kill a Stone Talus you must attack the Obsidian/Gem like outcropping on their back. The easiest way to do this to the simple Stone Taluses is to Climb onto their back and attack the growth with a Melee weapon. All weapons will work, but mining weapons like Hammer and Boulder Breaker are more effective in terms of damage.

Variant Taluses may be more difficult to climb, but you can get around it by neutralizing their elemental damage. Throwing bombs can also be moderately effective, but aiming them can be a bit difficult and finicky. If you have the resources Bomb arrows are also quite effective.

You can see an example of someone fighting a Stone Talus below:

For Elemental Taluses, shoot them with an arrow or hit them with a weapon of the opposite element (fire for ice and vice versa) before touching them. (Or equip iceproof/fireproof armor.)

Two cases:

  • For Taluses with the ore deposit on top of them, climb them, equip a two handed weapon and hold the attack button to keep spinning.

    If you have a mining weapon (Boulder Breaker, Iron Sledgehammer, Stone Smasher, Cobble Crusher), use it because you get a damage bonus multiplier:

    Iron Sledgehammer has a four times multiplier on Talus enemies. Boulder Breaker has a two times multiplier. The Stone Smasher and Cobble Crusher also have a two times multiplier.

  • For Taluses with the deposit on their back: two easy strategies.

    • Method 1: Climb them, then jump towards the deposit and keep shooting it in slow motion mode. Equipping a powerful bow and having stamina food is strongly recommended.

    • Method 2: If you have a spear, target the enemy and just keep hitting their deposit. They count as critical hits so the Talus will be stunned every time. Just be careful not to stand in its path. Repeat until it's dead.

Just as an addendum to Malco's answer, if you destroy the Talus' arms with bombs, the Talus will collapse and temporarily be stunned, allowing you to climb up and deal some damage to the gem on it's back.