Green - The green Chaos Emerald's power is the power of invisibility. In other words, it allows you to go around unseen by other people and undetected by any scouter.
Red - The red Chaos Emerald's power is the power of fire. It enables you to transform any attack into a "firey" form (example: Firey Kamehameha). When an attack is Firey, it's overall damage is increased. You also get surrounded in a cool looking aura of fire.
Blue - The blue Chaos Emerald's power is the power to transform into a water form and allows you to move around freely through cracks in the ground and you don't get hurt if someone steps on you while in this form. You can also jump into any body of water and automatically blend in, making it very hard for your opponent to spot you. Another possibility when you're in the water could be to create giant waves, even a tsunami...
Yellow - The yellow Chaos Emerald's power is the power of electricity. It enables you to spawn huge lightning storms and shoot powerful bolts of lightning from your hands. It can also enhance beam attacks like the Red Emerald (Example: Electric Kamehameha)
Purple - The purple Chaos Emerald's power is the power of True Sight. This enables you to where anyone is even if they are invisible (but it does not act like a scouter and give you the other person's pl). If you have a scouter, Dragon radar, or Helmet Sensor, this emerald enhances it by decreasing the search for Dragon Balls by 1 day.
Black - The black Chaos Emerald's power is the power of darkness. This enables you to surround everything in pure darkness, causing everybody to not see anything around them. Like the Red and Yellow Emeralds, this Emerald can enhance attacks too (Example: Dark Kamehameha)
Orange - The orange Chaos Emerald's power is the power of light. This enables you to counter the power of the Black Emerald.

All of the Chaos Emeralds enhance your speed and strength in battle. If one person has all of the Chaos Emeralds in his/her possession, that person can then ascend to the next level of power for a short period of time during battle (ex: if you have the ability to go Super Saiyan, you can go Super Saiyan 2 temporarily during battle). Also...try combining the powers of the Enhancing Emeralds to make your attacks even stronger than before.