Type refers to the way the dishwasher is built into the cabinet.
Freestanding, with a worktop. This sort can generally be built-in as well as the worktop can be removed.
Built-in – might not have top or side panels; must be enclosed and anchored to prevent it from tipping forward when fully loaded and dish racks are pulled out.
Semi-integrated – a built-in type that allows you to fit a panel to the door under the control fascia to match the rest of your kitchen cupboard doors.
Fully integrated – built-in with the whole front matching your kitchen and the controls inside the door.
Compact refers to the height, these tend to be built in (like an oven) or benchtop dishwashers. Slimline refer to a full height dishwasher, but they tend to be a good deal narrower to suit narrow margin installs.