Are you doing this DIY or having a cabinet maker build for you? Here is a link to a relatively easy DIY project. You just nail the support structure into the studs in your wall and slip the floating shelf over.

If you are having a cabinet maker do it, I think their process is a bit more involved. I think the guy that we talked to mentioned that they actually install supporting brackets behind your drywall or something.

Here is a link that might be useful: DIY Floating Shelves

Our cabinet maker is installing the shelves for us. When I told him I didn't want visible brackets he seemed unsure about it. But he is assuring me there will be no brackets. I just want to make sure they get installed properly. His work is excellent but I come from a small town so the open shelf concept might be a bit foreign to him.

We have a floating shelf over our buffet. Our furniture maker (extremely talented, BTW) wasn't too sure we could install it as a floating shelf, with enough support (?), but DH and our friend (a contractor) did a terrific job. They measured (measured, measured!) the distance between the studs in the wall, and marked the wall and the edge of the shelf accordingly. Then they drove (drilled?) large lag bolts into the studs in the wall and cut the heads off the bolts, drilled holes (sized to be a snug fit with the lag bolts) in the shelf using a drill press, and then slid the headless bolts into the holes they'd drilled into the shelf. Voila! Of course, our shelf size necessitated a very sound support system--your kitchen shelves might not be as heavy or as long.

I was thrilled, because I knew I would be happy with it installed with brackets, and really wanted it to be a floating shelf. Here's a pic (an old one):

The short description by cat mom is accurate.

To generalize this:
A series of rods sticking out of the wall go into holes perfectly aligned to receive them.
1./ Any metal rod sticking out of the wall, made any way.
2./ Any very well aligned hole in your chosen shelf material

Here is an example from five years ago. Jackww had metal rods welded onto a flat bar. Then he bolted the flat bar into the wall. This is one way to get bolts into the wall studs. Probably he grooved out the backside of the shelf so that the flat bar would be hidden. His shelves were massively heavy oak used normally as countertop material.
See http://www.ikeafans.com/galleries/index.php?n=803
and http://www.ikeafans.com/forums/yes-s-ikea/5498-masterpiece-abstrakt-white.html

The strongest method is Metal rods.

Other methods work too. They all involve attaching something to the wall which then gets covered by the shelf after the shelf gets slid over it or onto it. This is why most floating shelves are thick.

In the Ikea web site and in Ikea stores, see Lack and Besta floating shelves. There are others too.

Here is a link that might be useful: LACK floating shelf for $15 at Ikea

Thank you for each of your responses. You have been a tremendous help. I do have another question about the process. We are tiling the whole wall where the shelves will be. Do we tile around the shelves after they are installed or tile first and drill through the tile? It seems to me it would be better to tile around the shelves but not sure.

I would drill and insert the metal rods, tile the wall and then slip shelf on. This should look fine as long as wall isn't wavy. If it is tiling after shelves are up would cover up the gaps.