walkindown33 - As PoliticallyIncorrect's picture shows, it's a tube with a hole. The purpose of it is to provide a comfortable mouth piece that protects your lips from the heat of the atomizer, while providing the convenience of being able to drip right into it without removing anything.
jctgh - The bridge isn't necessary for the atomizer to work well, but it IS necessary for using a cart. The bridge is what wicks the liquid down from the filler in the cart. So once you remove the bridge, that atomizer can only be used for dripping.
But, once you remove the bridge, it increases performance in several ways.
1. Less wasted juice, since the bridge isn't absorbing it. It takes 4-8 drops to prime an intact 510 atomizer. It only takes 2 or 3 drops to prime a bridgeless 510 atomizer.
2. More airflow to the atomizer, resulting in warmer vapor, and more of it. The "warmer" thing is what REALLY sold me. My main gripe with atties had always been that the vapor was too cold. Removing the bridge fixed that.
3. For me personally, it seems that because of the exposed coil with increased airflow, they are easier to clean completely and less likely to die from "heat death" when dry-burning them.
4. They perform more consistently for me, since the juice goes right on the coil immediately, rather than getting caught up in the bridge.
5. It's tastier. Since it's easier to get the atomizer squeeky clean, and there's nothing but the coil in there, you're getting the purest juice taste humanly possible.
Here is a great video that shows how to remove the bridge.
YouTube - Removing The Bridge