A regular electric oven uses a large coiled resistor as a heating element. A large electric current is passed through this element which generates heat, similar to the tungsten filament of an incandescent light bulb.
An induction stove doesn't actually generate heat itself, but rather induces it the pot or pan. It does this using a rapidly oscillating magnetic field. This field induces an electric current in a ferromagnetic pan which generates heat, thus heating the food.
Induction stoves are faster, safer, and more energy efficient than traditional electric stoves. It requires significantly less electricity to create the magnetic field than it does to drive enough current through a resistor to make it heat up. They are safer because they don't actually generate heat themselves. After taking a pot off the "burner" (glass surface) it quickly begins to cool.
With regards to differences in the cooking process, there's not much. You just need to be aware that it heats up much more quickly and cools down much more quickly than a traditional electric stove. It does affect your choice of pans however. All aluminum pans will not work, nor will some lower quality stainless steel ones. If a magnet sticks to the pan then it will work just fine. The pan must also have a flat bottom to make full contact with the surface.
I will skip everything that does not work on an induction range, as it is pretty obvious when you purchase cookware whether it is marked as suitable for induction or not.
The benefit of induction is an immediate start and stop of energy transfer, from the point of view of the cook. There are other benefits, like safety, but we focus on cooking here. For a two-hour stew, it really doesn't matter whether you use induction or anything else.
Material-independent
Your cookware should really, really match the size of the induction plates, as all induction stoves check the flow of current and disable the induction plate, if there is something wrong. You can use larger cookware, but this increases the uneven heating.
So, buying a full set of cookware is often a bad idea, you will always end up with something that doesn't quite fit. So triple-check the diameter of your induction plates with the cookware. (Also, you always end up with something that you never use in such a set and with one particular pan that you always use and that is always dirty when you need it. Match the cookware with your cooking habits and servings.)
Stainless steel
For stainless steel, there are three available bottoms. The least expensive stainless steel products will be usually made out of type 3xx series with nickel, have no core at all and will not work on induction anyway. Cookware out of 4xx series stainless steel will work on induction, but this cookware is not as resistant to corrosion. I do not recommend either cookware.
The mid-range items will have a noticeable disk attached to the bottom of the cookware. This is a layer of magnetic steel and a slice of aluminium. Some manufacturers put a thin-layer of copper in there too. This is just marketing and has no noticeable effect. If it's a noticeable disk on the bottom, avoid it - it will work, maybe you won't even notice anything bad, but you will lose the benefits of the induction.
Then you have the standard sandwich bottoms. The inner layer is full 18/10 stainless steel, while the outermost layer is 18/0 magnetizable stainless steel and the middle layers are made out of aluminium, to distribute the heat. This is a sandwich bottom with an aluminium core. Again, some manufacturers will declare their core as being made out of 5 or more layers of aluminium, but this is again just marketing. Aluminium core is okay. This is the choice if you don't want to waste money, but do need to be price-conscious.
Finally, you have sandwich bottoms with a copper core. They are like the standard sandwich bottoms, but with a copper core between two aluminium layers. The mean thing is though now - manufacturers are smart, so they often make a standard sandwich bottom with a tiny layer of copper. This is not the same as a real copper core though.
If you want to reap most of the benefits of an induction stove and want stainless steel products, you need to get something with a copper core. Again, it is completely fine to get other stainless steel products that are suitable for induction cooking.
Cast-iron
Most cast-iron cookware producers will warn about scratching, because cast-iron will never be as flat as a steel/aluminium bottom. If you are worried and must slide the cast-iron cookware on an induction stove, there is a simple solution: parchment paper between the stove and the pan.
Otherwise, there is nothing special about cast-iron on induction.
Best Answer
Most reputable sources say that curved surfaces such as woks don't work as well on induction stoves. They even make special tools and cooktops for inductively heating woks. This phenomenum could be because of the angled surface or the extra distance from the cooktop, but it's probably both. It's not that these surfaces are immune to induction heating, just that they aren't efficient.
Angled surfaces won't respond as well to induction stoves since the maximum amount of energy is generated in a surface when it's flat with respect to the stove. (Magnetic induction is proportional to the cosine of the angle between loops, for those of you who want to know the physics).
Distance is probably the biggest factor. Induction works at a small distance from the stove (there are numerous examples on the web of induction heaters working through newspapers or a magazine.) However, the design of these cooktops makes it so the magnetic field that drives the heating dies off over a very short distance (For example, the Swiss Federal Office of Public Health determined when studying their effect on pacemakers that the field is negligible after 10 cm or 4 in).
As far as what materials work, usually cookware that's induction cooktop compatible has a curly icon on the box to indicate this. Common compatible materials include magnetic stainless steels (not austenitic steels), iron, and carbon steel. The article on induction cooking on Wikipedia has a great explanation on what cookware works for induction cooking and why.
There common rule of thumb is that if a magnet sticks to the bottom of a pot or pan, it's compatible with an induction stove. Surface resistance is a more scientific measure, but to get the complete picture of why certain things do or don't work would require a college course in material electromagnetics. For a more detailed analysis, see the wikipedia page I linked. For most cookware these are only minor concerns compared to the magnet test, until you accidentally leave aluminum foil on an induction stove and it starts to melt.
It should be noted that cookware needs some inductive metal in it, but not all of it needs to be an inductive metal. Some cookware designed for an inductive stove has an iron bottom with aluminum sides because aluminum conducts the heat better for more even cooking. Clad cookware with a mixture of the right metals works too.