Pokémon Labs are the places where Pokémon Professors (such as Oak, Elm, Birch, Rowan, Juniper, or Sycamore) give the Trainer a First partner Pokémon in the beginning of the story. The Labs are the places where the Professors do their research, the research area varies between the Professor in question's area of interest.
Professor Oak is featured in Pokémon Red, Blue and their remakes, FireRed and LeafGreen. In these games, Oak either gives the Trainer a Bulbasaur, a Charmander or a Squirtle, but in Pokémon Yellow he gives the Trainer a Pikachu.
Professor Elm is featured in Pokémon Gold, Silver, Crystal and their remakes, HeartGold and SoulSilver. In these games, Elm either gives a Chikorita, a Cyndaquil or a Totodile to the Trainer. This Pokémon lab is in New Bark Town.
Professor Birch is featured in Pokémon Ruby, Sapphire and Emerald. In these games, Birch either gives the Trainer a Treecko, a Torchic or a Mudkip.
Professor Rowan is featured in Pokémon Diamond, Pearl and Platinum. In these games, Rowan either gives the Trainer a Turtwig, a Chimchar, or a Piplup.
Professor Juniper is featured in Pokémon Black and White, Black 2 and White 2. In these games, Juniper either gives the Trainer a Snivy, a Tepig or an Oshawott.
Professor Sycamore is featured in Pokémon X and Pokémon Y. In these games, Sycamore either gives the player a Chespin, a Fennekin or a Froakie. He later gives the player a Bulbasaur, a Charmander or a Squirtle.
Each Generation has three different starter Pokémon, most of whom are Grass, Fire or Water-type Pokémon.