Every three hours, Jessie and James take over two maps: one early on and one further along. On their maps, half the defenders become Team Rocket's Pokémon (Ekans, Koffing, Meowth, Zubat, Grimer, Wobbuffet, Houndoom and more), with more levels and more health and damage than the map's own. Map info and the map list show where they are.
The stash
Your first win on one of their maps each time they take over pays Team Rocket's stash: three times the win's money, and sometimes an Item Box. You can take a stash up to eight times a day.
The hideout
After taking their stash 5 times, you find the Team Rocket Hideout (in Kanto, after Celadon Gym). Giovanni waits there with his team, and the hideout has 15 floors to climb like a gym's.
Missions reward beating Team Rocket: see Quests and missions.