Instructions
Head to the Route 208 tunnel. Use a Pokemon who knows the Surf, Rock Smash, Climb and Strength moves in order to climb to the top of Mt Coronet through the tunnel.
Exit the tunnel and you'll find yourself at the top of Mt Coronet. Jupiter and Mars, two Team Rocket Commanders, will be waiting for you. They engage you in a battle and your Rival will appear to help you with the double battle. After you defeat the duo, your Rival heals any damage your Pokemon withstands.
Fight Cyrus after your Pokemon is healed. He is someone who is obsessed with catching Palkia. After he is defeated, you will see Palkia.
Fight Palkia by hitting it with strong attacks. Use moves such as Hypnosis or Stun Spore. These moves will send it to sleep or paralyze it. After it becomes weak enough, usually down to only 1 or 2 HP, start throwing Ultra Balls at it to capture it. If you are lucky to have one, you can use a Master Ball, which captures any Pokemon every time.