After you complete the main game, you can capture the four Tapu of the Alola Region.
Catch the Other Tapu
After you become the Champion, you can go to the ruins on each of the four islands and touch the statue to battle the Tapu of that island, for a chance to capture it. This page explains where the ruins are and how to solve the block puzzles in each one.
Capturing Advice
The Tapu are difficult to catch, so bring a Pokémon that knows False Swipe so you can bring the Tapu's HP down to 1, and bring a Pokémon that can inflict sleep. Then you should get Dusk Balls and Quick Balls from the Pokémon Center on Route 8, Timer Balls from the Pokémon Center in Paniola Town, and Ultra Balls from any Pokémon Center.
You can try throwing a Quick Ball at the start of the battle, although it has a low chance of success. Then after lowering the Tapu's HP as much as possible and inflicting sleep (or another status ailment), throw Dusk Balls (the battles take place indoors, so the Dusk Ball bonus will be in effect). If the battle has gone on for a long time, start throwing Timer Balls instead of Dusk Balls. If you run out of Timer Balls and Dusk Balls, throw Ultra Balls.
If you KO the Tapu
If you KO the Tapu instead of catching it, you can get another chance by battling the Pokémon League again. Because of that, you might want to try catching all four Tapu, and if you KO one or more of them, you can battle the Elite Four again to get another chance to catch them.
Ruins of Life
The Ruins of Life are on Akala Island, north of where you battled Olivia. To get there, go east from Konikoni City and go east through Memorial Hill and Akala Outskirts.
Inside the ruins, there is a block puzzle that you have to solve to reach the statue in the northern room of the ruins. If you want to know how to solve the puzzle, click the button.
After that, go north into the northern room of the ruins, go up the stairs, and touch the statue. If you have become Champion, Tapu Lele will come out. Tapu Lele's type is Psychic Fairy.
Ruins of Abundance
The Ruins of Abundance are on Ula'ula Island, north of Haina Desert, which is north of Route 13.
Inside the ruins, there is a block puzzle that you have to solve to reach the statue in the northern room of the ruins. If you want to know how to solve the puzzle, click the button.
After that, go north into the northern room of the ruins, go up the stairs, and touch the statue. If you have become Champion, Tapu Bulu will come out. Tapu Bulu's type is Grass Fairy.
After you are done, you will find that the rocks are blocking your exit. If you want to know how to solve the puzzle, click the button.
Ruins of Hope
The Ruins of Hope are on Poni Island, east of Poni Breaker Coast in the southeast of the island.
You will have already solved the block puzzle earlier during the story, so just go into the northern room of the ruins, go up the stairs, and touch the statue. Tapu Fini will come out. Tapu Fini's type is Water Fairy.