A-Class
This is a completely purpose built motorhome, the manufacturer builds the body onto a new chassis/engine unit.
The advantages are that the interior is more spacious because the 'cab' area is usually part of the accommodation, many have a 'pull-down' double bed above the front seats, this can of course be left made up - no more battles with the table & assorted cushions needed to make the bed!
Something that may put people off is that some older A-Class designs have no drivers and passenger doors, or only a drivers side door.

Coachbuilt
Sometimes known as C-Class. The manufacturer uses the chassis & cab of the base vehicle and builds a new body onto the back of it, these make up the largest proportion of motorhomes. The part of the body extending over the cab is known as the 'over-cab' or 'luton' and is often a sleeping area - though adults will need to check that here is sufficient space as headroom is often restricted. An alternative, called a 'low-profile' has a much smaller over-cab space that is used purely for storage purposes

Demountable
A dual purpose motorhome, a standard 'pick-up' type vehicle with a specially made body that fits onto the pickup bed. These have the advantage that you can 'demount' the body at a campsite and then use the pickup for day trips etc.

Van Conversion
These are conversions to a standard panel van, the manufacturer fits the windows and interior.
There are three types:
Fixed roof - No alteration are made to the van body, the advantage in the low height of some base vehicles means that it's often possible to get under those pesky height barriers! Disadvantage, unless you are very short, is that you may end up with a permanent stoop!
Elevating or rising roof - Also known as 'pop-top'. The manufacturer cuts away the van roof and fits a, usually fibreglass, roof section that can be raised up when the vehicle is stationary to gain increased headroom. Often extra bunks are accommodated in the roof space, although these may only be suitable for children. Has the advantages of the fixed roof conversion without the disadvantage!
High top - Similar to the rising roof conversion except that the manufacturer fits a permanent high roof section to the van.