The Mew quest is one out of many additional, not story related Legendary Pokémon quests and is the only way to catch the Mythical Pokémon Mew.
Accomplishment of the dex requirements
In order to move forward with the Mew quest, you need the owned data of all Kanto Pokémon excluding legendary Pokémon on your Pokédex.
You also need at least 71 different evolution data in your Pokédex. Spawns for all Pokémon can be found in their specific pages.
More specific information regarding owned and evolved Pokémon and their mechanics, in regards to the Dex, can be found at the Pokédex page.
The first step of this quest is to obtain the seen data of the Kanto legendary Pokémon.
The seen data of these Pokémon can be obtained by battling with them. They appear as overworld NPC’s and are one-time battles. Once you have fought them, the seen data is saved, no matter if you win or lose.
Articuno is located at Seafoam Islands B4F. You will need Surf or a surf mount to reach it.
Zapdos will be inside the last room in the Power Plant. You will need Surf or a surf mount to reach the Power Plant.
Moltres will be at the exit of Victory Road Kanto. The seen data for Moltres should have already been obtained while doing the Kanto story, since battling it is needed in order to pass through Victory Road Kanto.
Mewtwo will be inside Cerulean Cave B1F in Kanto. Mewtwo can only be reached after doing the Cerulean Cave quest.
Old Sea Map
Now, when all requirements have been met, speak to the Know it all Man inside the Celadon Office, in Celadon City. He will reward the Old Sea Map, which is needed in order to travel to Faraway Island.
With the help of the Old Sea Map, it is possible to travel to Far Away Island with a Lapras located in the bay south of Pallet Town. The travel will only be successful when the interaction to Lapras is done at 12:00-13:00 Pokétime.
Once arrived, a small way follows to the second part of the island where Mew hides at a bush.
An interaction with the bush will trigger the fight with Mew. Make sure you bring sufficient Pokéballs and your desired sync. After the fight is over, you can leave the island with the sailor located at the beach on Faraway Island.
In case you were unsuccessful in catching Mew, you can earn another chance at catching it by beating Nikola.
Legendary Pokémon Quests Megathread