If you're looking at kitchens that were designed in the last few years, you'll notice a definite trend towards undermount sinks. It's easy to see why they're popular: having a sink that attaches to the underside of the countertop, rather than sitting on the top, makes for a clean, minimal look. And you can brush crumbs directly into the sink, which is nice.
What isn't so nice: because of the way these sinks are installed, there's always going to be a tiny lip where the countertop meets the sink, and overlaps it just a bit. I have an undermount sink, and I speak from experience when I say that, if you're not careful, this little juncture can get really grimy and gross. But the good news for those of you obsessed with cleanliness, or just a hyper minimalist look, is that there is another option: a sink that's built right into the countertop, so there's no seam at all.
Integrated (sometimes also called integral) countertops and sinks come in all kinds of materials: stainless steel, stone, solid surface. (In the case of marble or granite, the sink is almost always a different piece that's seamed together with the countertop, but since there's no lip the effect is the same.) Granted, none of these options are particularly cheap — but if you were planning on going with one of these materials anyway, an integral sink is definitely worth considering.
With solid surface countertops, like this Corian one spotted on Sapphire Spaces, you can get not just an integrated sink but an integrated drainboard too. The look is very, very clean, with no visible seam at all between the sink and countertop.
As noted above, marble sinks are often made from a different piece of marble (or joined together from multiple pieces of marble) and then seamed together with the countertop. If done well (like in this example from Rue Magazine), the effect is very similar, as if the countertop and the sink are all one piece. (Also lead image above.)
If you look carefully at this marble sink from Harmony and Design, you'll notice that the sink itself is one piece of marble, with the adjoining countertops carefully lined up with it.
This kitchen from Poppytalk features another kind of integral sink, in stainless steel. The effect is a bit industrial, but the adjoining marble countertop adds a touch of warmth.
Here's another built-in stainless sink, in a kitchen from Inside Out. In this case, the backsplash is part of the countertop, too.
The apron-front soapstone sink in this kitchen from Woon Stijl is joined to the countertops in the same way as the marble sink above.
The stainless countertops and sink in this kitchen from House & Home coordinate nicely with the stainless appliances.
It's even possible to pour a concrete countertop with an integrated sink, although this is much more common in the bathroom (which is understandable, because bathroom sinks are shallower and the countertops tend to be much shorter). The example here, from Haus Concrete Surfaces, is certainly very striking — both minimal and luxurious.