What type of joint is most appropriate for portions of furniture that are likely to be screwed and unscrewed in order to move it around?
There are actually numerous options for this. The best choice(s) depend on the size and weight of the furniture, aesthetic concerns and the thickness and strength of the stock used.
At the most basic you can screw the furniture together, and when building in hardwood in particular screws can be driven home, backed out and screwed back in successfully a number of times. Larger, more coarsely-threaded screws (e.g. coach screws) tend to work better for this as long as their diameters are not too great for the thickness of the material used.
In addition to directly screwing into wood, various other threaded metal fasteners are available, the simplest being bolts locked in place with hexagonal or square nuts — usually the projecting end of the bolt and the nut are hidden on the underside or interior surfaces of pieces (as often done in modern bed construction).
Here is a montage of images showing knock-down fasteners of various kinds:
Source: Good Wood Joints by Albert Jackson & David Day.
There are also a few traditional joinery techniques that don't rely on glue but on interlocking joint surfaces or on a wooden locking mechanism.
The simplest form of locking joint in Western joinery is probably a wedged projecting tenon, also referred to as a tusk tenon. Two examples below:
A single wedge (clearly visible in the top example above) or two opposing-angle wedges can be used to lock this joint in place. Note: where a single wedge is used the shoulder of the socket in the tenon must be angled to match, where two wedges are use the socket is square to the edges.
Self-locking joints are known in Western joinery but are most famous today from Japanese carpentry. Regardless of origin they can be extraordinarily difficult to shape; even the simplest type presents a great challenge in accurate sawing and chiselling and should probably only be attempted by the experienced woodworker.