1.Under Cabinet Mount
Just as the title says, this type of range hoods is fixed to the bottom of a cabinet. It may duct directly out the back, may go up through the cabinet or, may go up a little and then out the back. You may lose some storage in the cabinet, but so what? - it's always messy above the hood and never easy to get access to anyway! Very few of this type of range hoods have an remote blower.
This type of vent hood is directly affixed to the wall and may have a decorative chimney, or soffit, to line up with the top of your cabinets or to "disappear" into your ceiling. It may have an internal or remote blower, and it may duct straight out the back, be ducted straight up, or up and then horizontall.
3.Under Cabinet Mount
This type of range hoods hangs from the ceiling over an island or peninsula. It may also hang from peninsula wall cabinets. You may choose to hang it higher to avoid blocking a view and, if so, it will need more coverage than it would in a wall mount situation. I would recommend at least an additional 3" all round (30" stove would require a 36" hood. It will also need more CFM to compensate for the extra height and for breezes passing through your kitchen.
4.Under Cabinet Mount
Venting downwards has come a long way in the last 20 years and is mostly used in kitchen island or peninsula situations. This type of unit can also be used for a gas cooker near a wall. But why? The fumes are drawn downwards under the floor and out (like for a clothes dryer) or into a basement and out. A lower powered model, suitable for an electric or small gas cooker, often has the power pack placed in the cabinet beneath. A higher powered unit is usually placed in the basement and thus avoids the loss of storage an in-cabinet model may cause in a kitchen cabinet. Of course, you may choose to put lower powered units in the basement too to avoid this loss of storage.
Downdraft units are relatively inefficient because they have to fight the natural upwards flow of air and are most often used in an island situation if you don't want to block a view - or if your range top needs to be placed in front of a window.
Some downdraft models run from the front to the back of the countertop and are mostly used between modular stove tops; others have their vents at the back edge of the stove top but will not fit whenever a full size range top is in use (unless the countertop is increased in depth).