In it's purest form, the typical box camera shutter consists of a trigger arm attached to a spring which itself is attached to a plate that covers the aperture. Pressing the shutter tensions the spring, which releases at its maximum tension point and pulls the plate to quickly uncover the aperture.
There are as many different variations on this concept as there were manufacturers. The shapes of all of the pieces vary greatly - probably to avoid patent problems. Features were added like bulb modes, different apertures, filters and even different shutter "speeds" on advanced models like the Zeiss Box Tengor.
Over time, two general categories of box camera shutter design emerged. There is probably an official name for these – but for the sake of this overview, I'll call them "Always On" and "Always Off" designs.
The "Always On" Shutter
The "Always On" shutter is the simplest, but also the oddest version for a modern photographer to get used to using. Most of us are used to pressing the shutter lever or button and having it spring back automatically. "Always On" shutters are... well... always on - meaning that the spring inside always holds enough tension to snap the shutter open. This means that pressing the lever down takes a photo and the lever stays down. It is, in fact, ready to take another photo immediately. You must then press UP to take the next photo.
"Always On" shutters were more common in earliest days of box camera history - but continued to be used on the least expensive cameras well into the 1940s and beyond.
The "Always Off" Shutter
Far more common are shutter levers that bounce back after you activate them. These "Always Off" shutters react the way we have come to expect with analog cameras in general. You press the shutter and it returns automatically to the resting or "off" state when you let go. "Always Off" shutters are slightly more complicated, requiring a few more parts internally, but were so effective that they became the norm – and are still used today in Holga and Diana cameras.
The slides below show a very common, very minimal "Always Off shutter design and what happens when you take a snap. Click each image to enlarge.
Figure 1: The base plate is usually one piece of thick metal that has been stamped with a die. The aperture hole is cut out and the three stops are three-sided cuts that are bent upwards along the uncut edge. The Shutter Wheel is nearly flush with the Base plate and is attached with the brass rivet at its pivot point. The Trigger Arm is also attached by a rivet with a tiny bit of standoff so that it covers the Shutter Wheel.
Figure 2: Pressing the Trigger Arm down transfers energy into the Trigger Spring. The shape of the Trigger Arm is carefully designed to both cover the aperture when necessary, and to transfer energy into the Shutter Spring.
Figure 3: As the Trigger Arm reaches the stop, the thinner, longer shutter spring fires first, spinning the shutter past the aperture. Brass rivets allow the Shutter Spring ends to rotate, flipping the spring's position to store energy for the return trip (very clever).
Figure 4: The Shutter Wheel Stop catches the spinning wheel, stopping the exposure and positioning the Shutter Spring to accept energy from the Trigger Spring, which is still storing energy.
Figure 5: When the user releases the trigger, the stored tension in the Trigger Spring releases, which again tensions the Shutter Spring. As the Trigger Arm returns to cover the aperture, the Shutter Spring flips and activates again, returning the Shutter Wheel. This also-very-clever bit of timing is essential to prevent a second exposure as the Exposure Window spins backward across the aperture.
There are of course many, many variations on this theme and even the shape and size of the various elements are different from camera to camera – but the basic principles and simple mechanics remain roughly the same to this day.