Chopped and channeled refers to two different things, but are usually done together to create a specific look usually utilized by hot rodders.
Here is a decent attempt at a Photoshop "chop job" (pun intended), but it is a pretty good representation of a before and after of what it would look like:

In this image you can see the top is lowered. The basic events would occur something like this:
- Glass is taken out of the car
- All pillars are marked to take the same amount out of the them
- A chop saw (or sawzall or grinding tool of some sort) is used to cut the pieces out
- The top is re-used, but may need to be modified, depending on how the pillars will go back together
- The pillars are reformed due to a difference in angles/slopes to allow the new transition of the roof to look natural
- Once all metal work is fabricated, everything is welded together
- Body work is done to make finish correct to include paint
- Glass is cut to fit the new height and re-installed in the vehicle
Channeling a vehicle where you cut out the floor pan of a vehicle and allow the frame to come up farther into the vehicle. The main purpose of a channel job is to allow the body of the car to sit closer to the ground without suspension travel. This process was done before when there wasn't as many options or manufacturing techniques available to modify the suspension in vehicles. It is a lot less common today to see this kind of work done. It is very involved and requires a lot of welding skills to accomplish.
The main reason someone would do either of these processes to a car is strictly for aesthetics. It's just the way the person wants a vehicle to look.
Negative effects for chopping the top are head room for the driver and passengers. To compensate, the seats can be lowered or lower seats added to accommodate. Doing this, though, makes it harder to see around the vehicle (line of sight). In all things there is a trade off.
As for channeling a vehicle, there needs to be extra attention give to clearance of the wheels to wheel wells as well as how far the front wheels turn. If not, rubbing will occur. In most of the older cars where this is applied, a stiffer suspension will probably be needed so as not to allow for the "boat" motion to occur. Too much bounce/jounce can cause tire contact with the body as well.
Basically there is no vehicle requirements to perform a chop job. It can be done to any vehicle as long as the person doing it has the skills to make it happen. Does this mean it would look good on any vehicle ... the simple truth is probably not, but that is up to personal interpretation.
The main thing you'd need to channel a vehicle is a frame separate from the body. Many cars today are unibody setups (the car body acts as the frame), so don't have a typical frame. In these cases, though, there usually are many options to lower the vehicle to the ride height which the owner would like.
In either case of Chopping or Channeling, these are usually applied to vehicles of American origin from the 1930's, 40's, & 50's. There is usually a specific look the owner is going for to recreate how it "used to be done" back in the old days. This doesn't mean the work isn't done to other cars outside that realm, though.