The original Pokédex contained 151 Pokémon species.
The number comes from the first-generation games, which introduced the Kanto region and species numbered 1 through 151. Mew, listed as number 151, was not normally obtainable through ordinary gameplay in the original releases and was distributed through special events.
Many people remember the first-generation total as 150 because Mew was hidden from normal play and because players encountered the 150 regular species first. The complete National Pokédex count for the original generation therefore includes Mew.
The first games encouraged trading because some Pokémon were exclusive to one version. Completing the original Pokédex required players to exchange creatures between Red, Green, and later Blue, as well as obtain event-distributed Pokémon such as Mew.