Sink styles have come a long way. A home builders or renovator is no longer restricted to installing basic a porcelain drop-in sink with a wide rim that rests on the edge of the counter cutout. A contemporary consumer-grade sink can be above the counter, below it and even a part of it. Two types of sink styles that do not stick up above the counter are "undermount" sinks and integral ones. These each have their advantages, but they also both have an impact on how you handle replacing them.
Undermount sinks are mounted literally under the counter. They don't have rims the fit over the top of the counter; instead the counter serves as the rim. These are also called bottom-mount sinks. Undermount sinks can be made from common sink materials like stainless steel or from molded composites. Cast iron is a little less common because of the weight. Home Depot notes that if you have a laminate countertop, an undermount sink will not be appropriate because of how the edges of the laminate countertop are fashioned.
Integral sinks are actually part of the counter and are the same material as the rest of the counter. This style might be a combination of components put together or it might be one long slab with a carved or molded section that is the sink. In either instance, the sink and counter appear to be a single piece. Integral sinks can be made from lots of different materials including stainless steel, cement and composites.
Undermount sinks don't have to be made of the same material as the rest of the counter, so if you have to replace just the sink you'll have more variety to choose from. Cleanup of an undermount sink and most integral ones is simple, because there's no rim or ridge to trap crumbs and other matter. The major disadvantages to an undermount sink are its seals and its shape. The sink and counter edges have to be connected and thoroughly sealed so water and debris can't get in there. IHS Contractors notes that if there are any gaps or crevices, mold and bacteria can grow and create a health problem. In addition to that, the countertop holes actually have to be custom-cut to fit the sink. If you have to replace the sink, you have to get one that is exactly the same size. That's one factor you can't change without replacing the entire countertop.
Integral sinks are fashionable, and they clean up well. Those that are a part of the same slab of material as the counter don't have the same issues with getting the seal right between the counter and sink top, which is one less thing for you to worry about. However, if you have to or want to replace the sink, you have to replace the counter. Integral sink and counter combinations made from separate pieces fused together are not be meant to be taken apart.