Auron's Celestial Weapon - Masamune
In order to acquire Auron's ultimate weapon the first thing you'll need to do is find the Rusty Sword that's just outside of the Caverns of Stolen Fayth. If you can't remember where that is, it's off the northeastern exit of the Calm Lands. Take a look at my guide that I linked to for pictures if you're still lost.
After you've acquired the Rusty Sword you'll need to take it to the Statue of Lord Mi'ihen which you can find in Mushroom Rock Road. To teach this statue take the airship to Djose Highroads and head west when you arrive up the crates and into the next area. Follow the path south a little until you see an alcove to the north.
In this alcove you'll find an elevator to ride up (as seen in my screenshot) to a platform which is where the Statue of Lord Mi'ihen can be found. Use the Rusty Sword to make a glyph appear on the wall and then inspect the glyph to use your Celestial Mirror to acquire the Masamune.
Once you've acquired the Masamune the next step is powering it up! In order to do this you'll need the Mars Crest and Mars Sigil. You can obtain the Mars Crest in bottom (Oldroad) section of Mi'ihen Highroad - you may already have it from the start of the game. The Mars Sigil can be obtained by completing 10 Species or Area Conquests at the Calm Lands Arena. The man here will give you the Mars Sigil once you've done that.
To get the Mars Crest you'll want to take the airship to Mi'ihen Highroad. Grab a chocobo while you're here to cut down on travel time and then head north until you reach the gate and the path splits. If you're having trouble finding this area use my screenshot below as a guide.
From this location you'll wanna cut back in the direction you came but this time take the eastern most road - the oldroad. You'll be on the lower road this time which you'll want to follow all the way to the end where you'll find a treasure chest that contains the Mars Crest.
If you're having trouble finding it take a look at my screenshot below! Also, if you didn't get the chests here on your last run through - you can reach them via Chocobo Feathers on the ground (much like how we got to Remiem Temple before). One of the chests contains a Fortune Sphere so it's definitely worth your time!
As aforementioned the Mars Sigil is obtained from the Calms Land Arena. You'll need to complete a total of 10 "Area Conquests" or "Species Conquests" and after you do, your reward will be the Mars Sigil. In order to complete either of these two objectives you'll need to purchase the weapons with the 'Capture' ability on them from the NPC at the Calm Lands Arena.
Area Conquest - You need to kill enough fiends to complete a whole 'area'. All of the areas and fiends you need can be found in my Monster Arena Guide.
Species Conquest - You'll need to kill at least one whole species of fiends. An example is killing all the element types or bird types.
Complete a total of 10 of these, it doesn't matter which of the two you do or how many of each, as long as you complete 10 total. Your reward will be the Mars Sigil.