An island fixture vent is the plumbing vent for the drain from a sink or dishwasher in a kitchen island. Plumbing drains require vents - a pipe that goes up from the drain to the air above - to keep water from siphoning out of the trap, that u-shaped bit of plumbing below your sink and other fixtures. The water in the trap is what keeps sewer gases and pests in the sewer system out of your home. The smaller pipes you see protruding from most homes' roofs are plumbing vents.
If a sink is installedagainst a wall, or in an attached peninsula counter, its vent simply runs up inside the wall. But vents from island fixtures can't do that (because there's no adjacent wall). An island vent typically runs under the floor to a nearby wall, then up and out the roof. They have a special design, and they sometimes require an opening in the nearby wall for access to a cleanout - a fixture that allows clogs to be removed.
All sinks and other fixtures with drains require a vent; your local plumber should be able to clarify your local requirements.
