The thing you're talking about is called an air gap. You can avoid an air gap by putting a high loop in the drain hose but there are certain locales (such as California, I think) that require an air gap in their code and inspectors generally enforce this. Some people put the air gap in to pass inspection and then replace it with a soap dispenser after inspection.

I live in CA so I did a bunch of research when I found out the rest of the country doesn't have to have that ugly, gunky air gap monstrosity on the sink. Result: it's all in the plumbing. The purpose of the air gap is to keep the dishwasher water from backflowing into your supply water. The air gap does the job, but so does a high loop in the plumbing. All things considered, I'd much rather have a high loop under the sink than a grody cylinder sitting on the sink. We have a hole for the air gap in case anybody ever wants to check, but we installed an air switch for the garbage disposal in that hole instead. In the picture below, you see the top of one of the push button switches (there are 3 to match your sink finish) at sink level. When you push the switch your garbage disposal turns on and off instead of a light switch look-alike on the wall.

Here is a link that might be useful: air switch

We had three holes drilled at our clean-up sink: one for the faucet, one for the air switch (GD) and one for the air gap (DW). Once our kitchen passes inspection we will most likely change out the air gap with a soap dispenser. Like suzanne, the research I have done on high loops show that it is as effective as an air gap.

Air gaps are required where I live too. They don't have to be in the sink or countertop - I've seen wall-mounted air gaps. Some dishwashers have engineering features that make an air gap superfluous.

We live in California too and got an exception to avoid the air gap because we have a dishwasher that does not require it. I believe at the time, Miele and Bosch were the only two that have installation guides that do not require this, but that may have changed since then. We do have a high loop in the drain as well, but the inspector didn't require that--basically just looked at the manual and signed off that we had installed it in keeping with the manufacturer's instructions even though this was in conflict with local code. However, I think different inspectors have different philosophies on this; ours was particularly progressive on several code issues we were pushing (CA lighting requirements, air gap, etc.) and basically signed off on everything we had a sound argument for. It's worth exploring, though. I much prefer the counter without the air gap to clean around (and we have a wall-mount faucet, so it would have been the only hole in the counter).

My CA city requires an airgap no matter what your dishwasher provides. I guess less thinking for them. :/

If you are planning on an inspection (I don't know if you're just replacing a counter or a whole kitchen), check with your local code enforcement and see if it's required and if so, will a particular DW will exempt you.