The Estus flask can be upgraded for more uses by Andre in Firelink. Upgrades require Estus Shards, which are found in various locations throughout the world.
You start the game with three uses in your Estus Flask. The first item you find is the Ashen Estus flask (which has one use). You can toggle the distribution of uses between the Estus flasks at Blacksmith Andre in Firelink Shrine. There are a total of 15 uses to be had if you find all 11 Estus Shards.
1) Proceed from the Tower on the Wall bonfire in High Wall of Lothric out onto a roof and then down a ladder. You can enter a building from this platform and eventually reach a large room with several enemies- melee guys and dogs. The Estus shard is on an anvil:
2) Proceed from the Undead Settlement bonfire through a building and into an open area with numerous enemies. Several will be gathered around a tree with a fire burning at its base. The shard is in plain view at the base of this tree:
3) Not far from the Crucifixion Woods bonfire in Road of Sacrifices, head between two stone buildings and under an arch of sorts. You'll see a fire burning on the ground. Head left and the shard will be on a corpse:
4) Close to the Cleansing Chapel bonfire. The shard is on a corpse at the base of a monument where four enemies are praying:
5) From Farron Keep bonfire in the Road of Sacrifices, look out and slightly left across the swamp and you'll see a broken bridge and an item under an arch, which is the shard on a corpse:
6) From the Anor Londo bonfire in Irithyll of the Boreal Valley, head up the stairs and straight ahead into the building. The shard is in a chest in a dimly lit area along the left wall:
7) From the Old King's Antechamber bonfire in the Catacombs of Carthus, there is an illusory wall across the room. Break it and take your first left (watch the slimes on the ceiling) and the shard is on a corpse at the end of the hallway:
8) From the Irithyll Dungeon bonfire, go across a bridge to the other side of the dungeon and downstairs, then outside to a giant open area. The path leads to the right and down some steps. In the next room is a chest- a mimic- that contains the shard:
NOTE: throw an Undead Hunter Charm at the mimic before you attack and it will not go hostile to you.
9) From the Dancer of Boreal Valley bonfire in High Wall of Lothric, go up the ladder and turn to the left. Continue along until you reach a lift that leads to the Consumed King's Garden. Jump off of halfway down, take the path to the right, and the shard is on a corpse:
10) From the Grand Archives bonfire in Lothric Castle, take the lift up and head out towards in the direction of the Prince Lothric boss area. As soon as you get outside, turn left and head up some stairs, circling around onto the roof where three big flying Knights will engage you. This circular roof has several exits that connect to a path around the perimeter, but one exit leads to an enclosed area where the shard is on a corpse:
11) Buy the 20,000 souls key from the Handmaid in Firelink that can be used to unlock the door at the back/right side of Firelink. Head up to the bridge and drop off the left side onto the roof. At the front of the building, drop down a level and go inside. The shard is on a corpse on a rafter:
Note: the first 10 are listed largely in discovery order, but this last shard can be gotten as soon as you can afford to purchase the key from the Handmaid.
On an aside: I didn't track Estus shard locations on my first playthrough, where I was very diligent with exploration and actually found them all. I decided that during my 2nd playthrough I would find them again and record the clips above, and when I reached the end of my playthrough, I had found 11 and thought that I was missing four. Since the flask maxes out at 15, I just assumed that there were 15 shards to be found. I couldn't understand how I had missed four of them and went crazy for a few hours scouring the world in the hopes of finding shards that didn't even exist. Then I remembered that you start with three uses in your flask and thought I was missing one shard, so I went crazy again looking for the one that I thought I had missed. I finally realized that the first item you pick up (the Ashen Estus flask) gives you a charge and that there are only 11 shards to be found.