Ignoring for a moment that text looks like it's meant for Dark Souls 3... Alright, let's see if I can remember how many special one-off or enemy drop only items are in the game from beginning to end. Or just weapons, I guess? Technically, rings might count, too... I'll narrow it down to "there's no way I would've guessed that without a guide" ones.
Demon Greathammer: Kill the Asylum Demon in Undead Asylum the first time you encounter it instead of the second.
Black Knight Sword, Black Knight Greatsword, Black Knight Halberd, Black Knight Axe, Black Knight Shield: Rare drops from Black Knight enemies. Each weapon only drops from the knights who use it, but the shield can drop from any knight. These enemies don't respawn until the very last area of the game, so if you want one of these weapons early consider leaving them be and coming back with 30 Humanity and a Covetous Gold Serpent Ring.
Demon Greataxe: Rare drop from Tauros Demons, including the boss in Undead Burg.
Drake Sword: Cut the tail off the Hellkite Drake on the bridge before Undead Parish.
Fang Boar Helm: Rare drop from the giant armored boar enemies. Armored boars are non-respawning and the drop rate for this helm is a bit low, but it becomes a guaranteed drop from the last boar in the game.
Blacksmith's Hammer: Drops from Andre of Astora in Undead Parish when killed. NOT RECOMMENDED since this permanently removes one of the most important blacksmiths in the game, and also because the hammer itself is a pretty awful weapon.
Channeler's Trident: Very rare drop from the blue-robed channelers. If you want this, expect to farm for a very long time.
Gargoyle Tail Axe: Obtained by cutting off the tail from the first gargoyle boss in Undead Parish. Can be acquired later in Anor Londo from the two gargoyle enemies.
Gargoyle Helm: Rare drop from gargoyles, including the boss in Undead Parish.
Gargoyle Halberd: Rare drop from gargoyles, including the boss in Undead Parish.
Demon Great Machete: Rare drop from Capra Demons, including the boss in Lower Undead Burg.
Sack: Drops from one of the butcher enemies in the Depths.
Spider Shield: Found on a corpse in the Depths. Drop down a hole behind the first butcher to reach it.
Dragon King Greataxe: Cut off the Gaping Dragon's tail in the Depths to get this axe.
Large Club: Rare drop from club-wielding Infested Barbarians in Blighttown.
Power Within: Found underneath the parasitic wall hugger in Blighttown.
Server: From the bonfire in the sewer at the bottom of Blighttown, cross the swamp towards the trees where all the giant leeches are. Near the back is a corpse with the Server on it.
Dragon Greatsword: Cut the Stone Dragon's tail in Ash Lake to get this.
titanite Catch Pole: Rare drop from titanite Demon enemies.
Astora Straight Sword: Found in the Valley of Drakes in front of an Undead Dragon.
Man-Serpent Greatsword: Rare drop from male Man-Serpents in Sen's Fortress and Duke's Archives.
Flamberge: Rare drop from female serpents in Sen's Fortress and Duke's Archives.
Symbol of Avarice: Rare drop from mimics. Guaranteed drop from the last mimic in the game. This drop can be rolled for repeatedly by throwing Lloyd's Talismans at a mimic. Wait until the previous talisman wears off before you try it again.
Demon Spear: Rare drop from the bat-winged demon enemies in Anor Londo.
Giant Halberd, Giant Greatshield: Rare drops from the giant sentinel enemies in Anor Londo.
Dragonslayer Greatbow: Found by jumping out a broken window near the top of the room before the boss in Anor Londo.
Silver Knight Straight Sword, Silver Knight Spear, Silver Knight Shield: Rare drops from Silver Knight enemies in Anor Londo.
Dragon Tooth, Havel's Greatshield: Found behind an illusory fireplace in the hallway with the second bonfire of Anor Londo.
Giant Blacksmith Hammer: Kill the Giant Blacksmith in Anor Londo. NOT RECOMMENDED, since this permanently removes an important blacksmith from the game.
Bloodshield: Found in the Painted World in front of an Undead Dragon.
Velka's Rapier: Found in the Painted World, through the locked door to the right of the phalanx enemies.
Notched Whip: Dropped by the NPC invader in the Painted World. He invades at the graveyard area to the right just after the giant double doors.
Priscilla's Dagger: Cut off Priscilla's tail in the Painted World.
Hammer of Vamos: Kill Blacksmith Vamos in the Catacombs for this hammer. NOT RECOMMENDED since this will permanently remove a blacksmith from the game. It's also a pretty terrible weapon.
Royal Helm: Obtained by killing Blacksmith Vamos. NOT RECOMMENDED since this permanently removes an important blacksmith from the game.
Gravelord Sword: Found by joining the Gravelord Servants covenant in the Catacombs. Climb into the open coffin before the titanite Demon with at least one Eye of Death in your inventory.
Moonlight Greatsword: Cut off Seath's tail in the Cystal Cave.
Tin Crystallization Catalyst: When Big Hat Logan moves to the Duke's Archives, buy all his spells, leave the area, and return. In the room where Seath is first encountered and couldn't be killed, find and kill Logan for this sorcery catalyst.
Jagged Ghost Blade: Rare drop from the ghost enemies in New Londo.
Ghost Blade: Rare drop from the screaming ghost enemies in New Londo.
Guardian's Tail: Cut off the Sanctuary Guardian's tail at the beginning of the DLC. This can be attempted again by fighting the two respawning Sanctuary Guardian enemies who appear in this boss room after killing Manus.
Oolacile Catalyst: Rare drop from the sorcerer bloathead enemies in Oolacile Township.
Obsidian Greatsword: Cut off Kalameet's tail.
Pursuers: Give the Soul of Manus to Snuggly the Crow in the Undead Asylum.
Sunlight Spear: In NG+, offer the Soul of Gwyn to the Sunlight Altar after reaching rank +1 in the Warriors of Sunlight covenant.