Double Battles are a feature that were introduced in Ruby & Sapphire. They allow for you to battle 2 Pokémon at once with 2 Pokémon which really adds to the strategy element of the game. However, Diamond & Pearl add to that strategy

In certain areas of the game, you will be teamed up with another trainer who will follow you and battle with you inside the certain area. There are a number of areas that do this. However while in the grass, you will still get challenged by Pokémon. However these will come in 2s and you and your partner trainer will battle them. While there are 2 Wild Pokémon out, you are unable to throw any Pokéballs, but if you defeat one of them, you will be able to capture the other. But be warned as the trainer you are with may inadvertantly defeat the Pokemon you are trying to catch.

Below are the places where you have these Team Ups and the Double Wild Battles:

Eterna Forest

In your first time in Eterna Forest, you will meet a girl who would like your help to get to the other side, which coincidentally is what you need to do so she joins you and helps you battle the trainers in here. She has a Chansey on her:

Wayward Cave

After hunting through the Wayward Cave on Route 206, you will come across a girl who needs help to get through. Just help her through the cave and you will eventually help her get out. She has a Kadabra on her:

Iron Island

In the cave on Iron Island, you will eventually run into a guy who looks somewhat similar to Aaron from the 8th movie. He requires your help to get to the secret exit of the cave. He has a Lucario on him. When you get him to the end of the cave, if you have a spare slot in your team, he will give you a Riolu egg, the pre-evolved form of Lucario. If you don't have a space, you can come back and get it when you do:

Victory Road

After you have obtained the National Dex, a new part of Victory Road opens up that takes you up to Route 224. When you first enter this area, it is covered in Fog which you must clear. Once you have, you will meet a girl who needs to get out of the area. She has an Arcanine:

Stark Mountain

After you have obtained the National Dex, when you get to Stark Mountain, you will meet up with a guy who wants to explore the Mountain and find the Volcanic Rock. Take him through the incredibly large cave and he will eventually take the Volcano Rock. You will need to go speak to him in his house in the Survival Area and go back to fight Heatran. He has a Claydol: