Players could collect 151 Pokémon species in the original Pokémon Red and Green games. These species formed what is now called the first generation of Pokémon.
The games included a Pokédex, an in-game encyclopedia that recorded the creatures encountered by the player. Completing it required more than simply playing one version, because Red and Green contained version-exclusive Pokémon. Players had to trade with someone using the other version.
The 151st Pokémon was Mew, a Mythical Pokémon that was not normally obtained through standard gameplay. Mew was added late in development and became the subject of rumors, helping sustain interest in the games. In ordinary discussions, the original roster is therefore often described as the 151 Pokémon of the Kanto Pokédex.
The number later grew with every new generation. Pokémon Gold and Silver added the Johto region and 100 more species, bringing the total to 251. That expansion is why 151 specifically identifies the original generation rather than the entire franchise.