As fruits ripen, a gas called ethylene (C2H4) is released. The effect is amplified as this gas stimulates ripening (positive feedback loop) and more ethylene production. Additionally, it will promote ripening of fruit nearby. Bananas and apples tend to release alot of this gas. You can take advantage of this in 2 ways.
1. Take a ripe banana and place it in a bag with unripe fruit. It could be hard avacados, green bananas, or any other unripe fruit. The gas stimulates the conversion of starch to sugars, making the fruit softer and sweeter. Keeping it in a warm closed bag will cause build up of ethylene, which will ripen fruit and produce more ethylene.
2. To slow down ripening of fruit, say you want to use fresh bananas in a pie next week, you can keep it well ventilated and away from ripe fruit.
But don't expect miracles, it still takes a while to ripen fully green bananas, but it can help. Also remember that the faster it ripens, the faster it will spoil. Use it while ripe before it rots!