An integral sink is made out of the same material as the countertop and is fused to it, creating a seamless transition that makes the sink and counter look like one unit. This option only exists with materials flexible enough to be fabricated into the shape of a sink basin, limiting you to quartz, stainless steel, glass, solid surfacing and other composite materials. (Glass, however, costs $200-$300 per square foot, so it is typically used for smaller surfaces such as vanity tops or dining bars, not the main kitchen counter.)
The seamless design makes cleaning a snap and, from a design perspective, makes the sink complete unobtrusive and allows the focus to remain on an elegant faucet or artistic backsplash. However, if the sink or countertop is damaged, you'll probably have to replace both items rather than simply swap out one or the other.