Team Mode is unlocked at lvl80. In this mode you can select a team to play together on game modes like Team Boss.
The two teams that exist are Team Rocket and Team Elite. You have the option to change teams if you want, but you will need to wait 7 days to join a new team.
In this Team Mode there's a Team Shop, Team Technology (similar to Talent Tree) and other game modes like Team Boss, Team GVG, Fishing, Cube Pokémon and Team Feeding House.
Team Boss being the main mode where being on the right team matters: If you're on the team that has the best players, your team should be able to clear more bosses, and win rewards according to each island that was defeated:
The rewards that you obtain from Team Boss are usually resources for Team Feeding House, Team Technology and Cube Pokémon.